I would say 1896, and could make arguments for 1796, but I con't have time for six paragraphs right now. 
1796? There was no Democratic Party in 1796. Just Republicans and Federalists, with the Republicans not the same party as today's GOP, but the ancestor of both of today's major parties.
I disagree with that analysis. Indeed there are traces of the Republicans of the 1790's in GOP, especially now with all the Southerners that have joined it but also even as far back as the founding of the party. However, the true ancestor both to the GOP and American Conservatism is the Federalist Party. A large number of Federalists joined the Republicans of Jefferson during the early part of the nineteenth century and they began to move the pary towards ideas of the Federalist Party. There was also the rise of the nationalists such as Henry Clay who shared the expansionist views of their party, but their economic platform was much more in line with the Federalist Party. That is why even as the Federalist were dying out, you had the Bank and tariffs passed in 1816. Jackson's movement and founding of the Democratic Party in the 1820's was a move to reclaim the the Republican's support of the common man, farmer and so forth against the elites. The name was adopted to emphasize that fact. That is why, at least until the last few decades, the Democrats were party of Jackson and Jefferson, and you still have Jefferson-Jackson dinners. I beleive the NY Democratic party organization dates itself back to 1792.
In the 1820's you had the old Federalists, nationalist Republicans, many of whom were from places that were not even states in the 1790's and a few other Republicans that came to believe the American System was the proper approach. They kept the name Republicans but altered it to reflect their nationalist views (National Republicans of course). Later they took the name Whigs as a means of forming an anti-Jackson coalition with several states-rights Democrats in the South. In the 1850's, the formation of the GOP was led by northern Whigs, and anti-slavery northern Democrats, they picked a name that would emphasize their goal (save the Republic), attach themselves to Jefferson's populist legacy even as their domestic policy was hardly anything close to what Jefferson would have supported (Banks, Tariffs and abolition) in his time and keep those anti-slavery Democrats on board at least until the issue of slavery was resolved.
Parties adapt with the times but typically they generally don't change as much as people would like to think. The Federalists got aced out of the game because of their appearence of elitist and aristocratic tendencies and various other actions, including their alleged treason. Successors thus did their best to expand their base to include a larger group of people to avoid that problem, while preserving the core desires and interest to serve, almost always to facilitate, support and protect commercial interests.
As for American Conservatism, the desire for limited goverment developed as a reaction to what was being done in the Progressive Era, during which time you had the Democrats moving towards the desire to use gov't to help the common man instead of seeing it as just a tool of the elites and seeking to restrain it like before. Prior to that era, most business interests saw Gov't as tool to advance their interests, but after it, such was mostly in the way. You still have a large number of Republicans today, particular establishment ones and those in the rural states who support subsidies to favored industries and of course pork barrel spending. That doesn't even get into the military and the spending there.
The best way to approach this issue is to back away from the how (as in how government is utilized) and focus more on the what and for whom it is being done. The how part of the question is variable based on the context and is subject to alteration out of political necessity, such as Conservatives and Libertarians moving toward each other during the Progressive and New Deal Periods and thus the shifting away from goverment in general, though still liking it on specific pet projects. Libertarians also trace a heritage to Jefferson, but rejected the embrace of Government and thus you had a split between classical and modern liberals.
I think most would accept that Jefferson is the father of American Liberalism, even without accepting all of the above. Fewer but still many, would say that Hamilton is the father of American Conservatism. John Adams, though not exactly friends of Hamilton, was hardly anything but a Conservative based on most definitions, save for a few of the modern ones, but especially in regards to the Burkean variety. He was certainly preoccupied with order, he supported a standing army and created the navy, was somewhat aristocractic, defended the British in Court after the Boston Massacre and was considerably on the religious side. I really don't want to go to twelve paragraphs on this, but this was before the Transcendentalist movement and its influences on the Unitarian church and thus while breaking away on matters of theology, they still possessed a similar worldview to that of the other, more Calvinist Congregational Churches in this era. Adams was the Conservative and Jefferson the Liberal in 1796. As for the parties, the easiest case can be made on the side of Jefferson's Party being the ancestors to the Democrats, because you have an organization tie, dinners that bare his name and the common objective, helping the little guy against the elites. On the side of the Federalist and the GOP, you have less, but enough. The early GOP's economic policy of Tariffs and financial stability was much the same as the Federalist's and for the same reason. Even with today's party they share the desire to serve business interests and preserve stability neccessary for them to thrive and aren't afraid (at least before the tea party era) of using government to do it when necessary.
