The Democratic Party presidential platform of 1860 was basically "see 1856 platform, ditto" so we'll lead off with the Republican platform of 1860 and follow with the two Democrat platforms.
The 1860 Republican platform consisted of 17 declarations, 7 of which directly addressed slavery. From The American Presidency Project at the University of California at Santa Barbara:
May 17, 1860Resolved, That we, the delegated representatives of the Republican electors of the United States in Convention assembled, in discharge of the duty we owe to our constituents and our country, unite in the following declarations:
1. That the history of the nation during the last four years, has fully established the propriety and necessity of the organization and perpetuation of the Republican party, and that the causes which called it into existence are permanent in their nature, and now, more than ever before, demand its peaceful and constitutional triumph.
2. That the maintenance of the principles promulgated in the Declaration of Independence and embodied in the Federal Constitution, "That all men are created equal; that they are endowed by their Creator with certain inalienable rights; that among these are life, liberty and the pursuit of happiness; that to secure these rights, governments are instituted among men, deriving their just powers from the consent of the governed," is essential to the preservation of our Republican institutions; and that the Federal Constitution, the Rights of the States, and the Union of the States must and shall be preserved.
3. That to the Union of the States this nation owes its unprecedented increase in population, its surprising development of material resources, its rapid augmentation of wealth, its happiness at home and its honor abroad; and we hold in abhorrence all schemes for disunion, come from whatever source they may. And we congratulate the country that no Republican member of Congress has uttered or countenanced the threats of disunion so often made by Democratic members, without rebuke and with applause from their political associates; and we denounce those threats of disunion, in case of a popular overthrow of their ascendency as denying the vital principles of a free government, and as an avowal of contemplated treason, which it is the imperative duty of an indignant people sternly to rebuke and forever silence.
4. That the maintenance inviolate of the rights of the states, and especially the right of each state to order and control its own domestic institutions according to its own judgment exclusively, is essential to that balance of powers on which the perfection and endurance of our political fabric depends; and we denounce the lawless invasion by armed force of the soil of any state or territory, no matter under what pretext, as among the gravest of crimes.
5. That the present Democratic Administration has far exceeded our worst apprehensions, in its measureless subserviency to the exactions of a sectional interest, as especially evinced in its desperate exertions to force the infamous Lecompton Constitution upon the protesting people of Kansas; in construing the personal relations between master and servant to involve an unqualified property in persons; in its attempted enforcement everywhere, on land and sea, through the intervention of Congress and of the Federal Courts of the extreme pretensions of a purely local interest; and in its general and unvarying abuse of the power intrusted to it by a confiding people.
6. That the people justly view with alarm the reckless extravagance which pervades every department of the Federal Government; that a return to rigid economy and accountability is indispensable to arrest the systematic plunder of the public treasury by favored partisans; while the recent startling developments of frauds and corruptions at the Federal metropolis, show that an entire change of administration is imperatively demanded.
7. That the new dogma that the Constitution, of its own force, carries slavery into any or all of the territories of the United States, is a dangerous political heresy, at variance with the explicit provisions of that instrument itself, with contemporaneous exposition, and with legislative and judicial precedent; is revolutionary in its tendency, and subversive of the peace and harmony of the country.
8. That the normal condition of all the territory of the United States is that of freedom: That, as our Republican fathers, when they had abolished slavery in all our national territory, ordained that "no persons should be deprived of life, liberty or property without due process of law," it becomes our duty, by legislation, whenever such legislation is necessary, to maintain this provision of the Constitution against all attempts to violate it; and we deny the authority of Congress, of a territorial legislature, or of any individuals, to give legal existence to slavery in any territory of the United States.
9. That we brand the recent reopening of the African slave trade, under the cover of our national flag, aided by perversions of judicial power, as a crime against humanity and a burning shame to our country and age; and we call upon Congress to take prompt and efficient measures for the total and final suppression of that execrable traffic
10. That in the recent vetoes, by their Federal Governors, of the acts of the legislatures of Kansas and Nebraska, prohibiting slavery in those territories, we find a practical illustration of the boasted Democratic principle of Non-Intervention and Popular Sovereignty, embodied in the Kansas-Nebraska Bill, and a demonstration of the deception and fraud involved therein.
11. That Kansas should, of right, be immediately admitted as a state under the Constitution recently formed and adopted by her people, and accepted by the House of Representatives.
12. That, while providing revenue for the support of the general government by duties upon imports, sound policy requires such an adjustment of these imports as to encourage the development of the industrial interests of the whole country; and we commend that policy of national exchanges, which secures to the workingmen liberal wages, to agriculture remunerative prices, to mechanics and manufacturers an adequate reward for their skill, labor, and enterprise, and to the nation commercial prosperity and independence.
13. That we protest against any sale or alienation to others of the public lands held by actual settlers, and against any view of the free-homestead policy which regards the settlers as paupers or suppliants for public bounty; and we demand the passage by Congress of the complete and satisfactory homestead measure which has already passed the House.
14. That the Republican party is opposed to any change in our naturalization laws or any state legislation by which the rights of citizens hitherto accorded to immigrants from foreign lands shall be abridged or impaired; and in favor of giving a full and efficient protection to the rights of all classes of citizens, whether native or naturalized, both at home and abroad.
15. That appropriations by Congress for river and harbor improvements of a national character, required for the accommodation and security of an existing commerce, are authorized by the Constitution, and justified by the obligation of Government to protect the lives and property of its citizens.
16. That a railroad to the Pacific Ocean is imperatively demanded by the interests of the whole country; that the federal government ought to render immediate and efficient aid in its construction; and that, as preliminary thereto, a daily overland mail should be promptly established.
17. Finally, having thus set forth our distinctive principles and views, we invite the co-operation of all citizens, however differing on other questions, who substantially agree with us in their affirmance and support.
APP Note: The Official Proceedings of the 1860 Republican National Convention indicates that the platform was announced and read on the second day of the convention (May 17, 1860).
And from the Democratic Party, 1860, again from the APP at UCSB:
June 18, 18601. Resolved, That we, the Democracy of the Union in Convention assembled, hereby declare our affirmance of the resolutions unanimously adopted and declared as a platform of principles by the Democratic Convention at Cincinnati, in the year 1856, believing that Democratic principles are unchangeable in their nature, when applied to the same subject matters; and we recommend, as the only further resolutions, the following:
2. Inasmuch as difference of opinion exists in the Democratic party as to the nature and extent of the powers of a Territorial Legislature, and as to the powers and duties of Congress, under the Constitution of the United States, over the institution of slavery within the Territories,
Resolved, That the Democratic party will abide by the decision of the Supreme Court of the United States upon these questions of Constitutional law.
3. Resolved, That it is the duty of the United States to afford ample and complete protection to all its citizens, whether at home or abroad, and whether native or foreign born.
4. Resolved, That one of the necessities of the age, in a military, commercial, and postal point of view, is speedy communication between the Atlantic and Pacific States; and the Democratic party pledge such Constitutional Government aid as will insure the construction of a Railroad to the Pacific coast, at the earliest practicable period.
5. Resolved, That the Democratic party are in favor of the acquisition of the Island of Cuba on such terms as shall be honorable to ourselves and just to Spain.
6. Resolved, That the enactments of the State Legislatures to defeat the faithful execution of the Fugitive Slave Law, are hostile in character, subversive of the Constitution, and revolutionary in their effect.
7. Resolved, That it is in accordance with the interpretation of the Cincinnati platform, that during the existence of the Territorial Governments the measure of restriction, whatever it may be, imposed by the Federal Constitution on the power of the Territorial Legislature over the subject of the domestic relations, as the same has been, or shall hereafter be finally determined by the Supreme Court of the United States, should be respected by all good citizens, and enforced with promptness and fidelity by every branch of the general government.
APP Note: The American Presidency Project used the first day of the national nominating convention as the "date" of this platform since the original document is undated.
And the 1856 Democratic Party Platform, referenced above as the 'Cincinnati platform':
June 02, 1856Resolved, That the American Democracy place their trust in the intelligence, the patriotism, and the discriminating justice of the American people.
 Resolved, That we regard this as a distinctive feature of our political creed, which we are proud to maintain before the world, as the great moral element in a form of government springing from and upheld by the popular will; and we contrast it with the creed and practice of Federalism, under whatever name or form, which seeks to palsy the will of the constituent, and which conceives no imposture too monstrous for the popular credulity.
 Resolved, therefore, That, entertaining these views, the Democratic party of this Union, through their Delegates assembled in a general Convention, coming together in a spirit of concord, of devotion to the doctrines and faith of a free representative government, and appealing to their fellow-citizens for the rectitude of their intentions, renew and re-assert before the American people, the declarations of principles avowed by them when, on former occasions in general Convention, they have presented their candidates for the popular suffrage.
1. That the Federal Government is one of limited power, derived solely from the Constitution; and the grants of power made therein ought to be strictly construed by all the departments and agents of the government; and that it is inexpedient and dangerous to exercise doubtful constitutional powers.
2. That the Constitution does not confer upon the General Government the power to commence and carry on a general system of internal improvements.
3. That the Constitution does not confer authority upon the Federal Government, directly or indirectly, to assume the debts of the several States, contracted for local and internal improvements, or other State purposes; nor would such assumption be just or expedient.
4. That justice and sound policy forbid the Federal Government to foster one branch of industry to the detriment of any other, or to cherish the interests of one portion to the injury of another portion of our common country; that every citizen and every section of the country has a right to demand and insist upon an equality of rights and privileges, and to complete and ample protection of persons and property from domestic violence or foreign aggression.
5. That it is the duty of every branch of the Government to enforce and practice the most rigid economy in conducting our public affairs, and that no more revenue ought to be raised than is required to defray the necessary expenses of the Government, and for the gradual but certain extinction of the public debt.
6. That the proceeds of the public lands ought to be sacredly applied to the national objects specified in the Constitution; and that we are opposed to any law for the distribution of such proceeds among the States, as alike inexpedient in policy and repugnant to the Constitution.
7. That Congress has no power to charter a national bank; that we believe such an institution one of deadly hostility to the best interests of the country, dangerous to our republican institutions and the liberties of the people, and calculated to place the business of the country within the control of a concentrated money power, and above the laws and the will of the people; and that the results of Democratic legislation in this and all other financial measures upon which issues have been made between the two political parties of the country, have demonstrated to candid and practical men of all parties, their soundness, safety, and utility, in all business pursuits.
8. That the separation of the moneys of the Government from banking institutions is indispensable for the safety of the funds of the Government and the rights of the people.
9. That we are decidedly opposed to taking from the President the qualified veto power, by which he is enabled, under restrictions and responsibilities amply sufficient to guard the public interests, to suspend the passage of a bill whose merits cannot secure the approval of two-thirds of the Senate and House of Representatives, until the judgment of the people can be obtained thereon, and which has saved the American people from the corrupt and tyrannical domination of the Bank of the United States, and from a corrupting system of general internal improvements.
10. That the liberal principles embodied by Jefferson in the Declaration of Independence, and sanctioned by the Constitution, which makes ours the land of liberty and the asylum of the oppressed of every nation, have ever been cardinal principles in the Democratic faith, and every attempt to abridge the privilege of becoming citizens and the owners of soil among us, ought to be resisted with the same spirit which swept the alien and sedition laws from our statute-books.
And Whereas, Since the foregoing declaration was uniformly adopted by our predecessors in National Conventions, an adverse political and religious test has been secretly organized by a party claiming to be exclusively American, it is proper that the American Democracy should clearly define its relation thereto, and declare its determined opposition to all secret political societies, by whatever name they may be called
 Resolved, That the foundation of this union of States having been laid in, and its prosperity, expansion, and pre-eminent example in free government, built upon entire freedom in matters of religious concernment, and no respect of person in regard to rank or place of birth; no party can justly be deemed national, constitutional, or in accordance with American principles, which bases its exclusive organization upon religious opinions and accidental birth-place. And hence a political crusade in the nineteenth century, and in the United States of America, against Catholic and foreign-born is neither justified by the past history or the future prospects of the country, nor in unison with the spirit of toleration and enlarged freedom which peculiarly distinguishes the American system of popular government.
Resolved, That we reiterate with renewed energy of purpose the well considered declarations of former Conventions upon the sectional issue of Domestic slavery, and concerning the reserved rights of the States.
1. That Congress has no power under the Constitution, to interfere with or control the domestic institutions of the several States, and that such States are the sole and proper judges of everything appertaining to their own affairs, not prohibited by the Constitution; that all efforts of the abolitionists, or others, made to induce Congress to interfere with questions of slavery, or to take incipient steps in relation thereto, are calculated to lead to the most alarming and dangerous consequences; and that all such efforts have an inevitable tendency to diminish the happiness of the people and endanger the stability and permanency of the Union, and ought not to be countenanced by any friend of our political institutions.
2. That the foregoing proposition covers, and was intended to embrace the whole subject of slavery agitation in Congress; and therefore, the Democratic party of the Union, standing on this national platform, will abide by and adhere to a faithful execution of the acts known as the compromise measures, settled by the Congress of 1850; "the act for reclaiming fugitives from service or labor," included; which act being designed to carry out an express provision of the Constitution, cannot, with fidelity thereto, be repealed, or so changed as to destroy or impair its efficiency.
3. That the Democratic party will resist all attempts at renewing, in Congress or out of it, the agitation of the slavery question under whatever shape or color the attempt may be made.
4. That the Democratic party will faithfully abide by and uphold, the principles laid down in the Kentucky and Virginia resolutions of 1798, and in the report of Mr. Madison to the Virginia Legislature in 1799; that it adopts those principles as constituting one of the main foundations of its political creed, and is resolved to carry them out in their obvious meaning and import.
And that we may more distinctly meet the issue on which a sectional party, subsisting exclusively on slavery agitation, now relies to test the fidelity of the people, North and South, to the Constitution and the Union—
1. Resolved, That claiming fellowship with, and desiring the co-operation of all who regard the preservation of the Union under the Constitution as the paramount issue—and repudiating all sectional parties and platforms concerning domestic slavery, which seek to embroil the States and incite to treason and armed resistance to law in the Territories; and whose avowed purposes, if consummated, must end in civil war and disunion, the American Democracy recognize and adopt the principles contained in the organic laws establishing the Territories of Kansas and Nebraska as embodying the only sound and safe solution of the "slavery question" upon which the great national idea of the people of this whole country can repose in its determined conservatism of the Union—NON-INTERFERENCE BY CONGRESS WITH SLAVERY IN STATE AND TERRITORY, OR IN THE DISTRICT OF COLUMBIA.
2. That this was the basis of the compromises of 1850 confirmed by both the Democratic and Whig parties in national Conventions—ratified by the people in the election of 1852, and rightly applied to the organization of Territories in 1854.
3. That by the uniform application of this Democratic principle to the organization of territories, and to the admission of new States, with or without domestic slavery, as they may elect—the equal rights, of all the States will be preserved intact; the original compacts of the Constitution maintained inviolate; and the perpetuity and expansion of this Union insured to its utmost capacity of embracing, in peace and harmony, every future American State that may be constituted or annexed, with a republican form of government.
Resolved, That we recognize the right of the people of all the Territories, including Kansas and Nebraska, acting through the legally and fairly expressed will of a majority of actual residents, and whenever the number of their inhabitants justifies it, to form a Constitution, with or without domestic slavery, and be admitted into the Union upon terms of perfect equality with the other States.
Resolved, Finally, That in view of the condition of popular institutions in the Old World (and the dangerous tendencies of sectional agitation, combined with the attempt to enforce civil and religious disabilities against the rights of acquiring and enjoying citizenship, in our own land) a high and sacred duty is devolved with increased responsibility upon the Democratic party of this country, as the party of the Union, to uphold and maintain the rights of every State, and thereby the Union of the States; and to sustain and advance among us constitutional liberty, by continuing to resist all monopolies and exclusive legislation for the benefit of the few, at the expense of the many, and by a vigilant and constant adherence to those principles and compromises of the Constitution, which are broad enough and strong enough to embrace and uphold the Union as it was, the Union as it is, and the Union as it shall be, in the full expansion of the energies and capacity of this great and progressive people.
1. Resolved, That there are questions connected with the foreign policy of this country, which are inferior to no domestic question whatever. The time has come for the people of the United States to declare themselves in favor of free seas and progressive free trade throughout the world, and, by solemn manifestations, to place their moral influence at the side of their successful example.
2. Resolved, That our geographical and political position with reference to the other States of this continent, no less than the interest of our commerce and the development of our growing power, requires that we should hold as sacred the principles involved in the Monroe Doctrine: their bearing and import admit of no misconstruction; they should be applied with unbending rigidity.
3. Resolved, That the great highway which nature, as well as the assent of the States most immediately interested in its maintenance, has marked out for a free communication between the Atlantic and the Pacific oceans, constitutes one of the most important achievements realized by the spirit of modern times and the unconquerable energy of our people. That result should be secured by a timely and efficient exertion of the control which we have the right to claim over it, and no power on earth should be suffered to impede or clog its progress by any interference with the relations it may suit our policy to establish between our government and the Governments of the States within whose dominions it lies. We can, under no circumstances, surrender our preponderance in the adjustment of all questions arising out of it.
4. Resolved, That, in view of so commanding an interest, the people of the United States cannot but sympathize with the efforts which are being made by the people of Central America to regenerate that portion of the continent which covers the passage across the Interoceanic Isthmus.
5. Resolved, That the Democratic party will expect of the next Administration that every proper effort be made to insure our ascendency in the Gulf of Mexico, and to maintain a permanent protection to the great outlets through which are emptied into its waters the products raised out of the soil and the commodities created by the industry of the people of our Western valleys and the Union at large.
Resolved, That the Democratic party recognizes the great importance, in a political and commercial point of view, of a safe and speedy communication, by military and postal roads, through our own territory, between the Atlantic and Pacific coasts of this Union, and that it is the duty of the Federal Government to exercise promptly all its constitutional power to the attainment of that object, thereby binding the Union of these States in indissoluble bonds, and opening to the rich commerce of Asia an overland transit from the Pacific to the Mississippi River, and the great lakes of the North.
Resolved, That the administration of Franklin Pierce has been true to the great interests of the country. In the face of the most determined opposition it has maintained the laws, enforced economy, fostered progress, and infused integrity and vigor into every department of the government at home. It has signally improved our treaty relations, extended the field of commercial enterprise, and vindicated the rights of American citizens abroad. It has asserted with eminent impartiality the just claims of every section, and has at all times been faithful to the Constitution. We therefore proclaim our unqualified approbation of its measures and its policy.
APP Note: The American Presidency Project used the first day of the national nominating convention as the "date" of this platform since the original document is undated.
For comparison, if interested
