The Democratic-Republican Party(Jefferson, Madison, Monroe) and the Whig Party (Clay, Harrison, Webster) were one of the two major parties of their time. D-R party was paired with the Federalists, and for a time was the only party during Monroe and JQ Adams. The Whigs were the opposition to the Jacksonian Democrats. Your examples were not third parties, they were the major ones. A better example you should have chosen was the Know Nothing Party. Even they had way more support then today’s fringe third parties.
Er, that was the point. They were major parties; they aren't any more.