A corrective footnote: Andrew Jackson didn’t found the Democratic-Republican party. That honor is usually assigned to Thomas Jefferson. He did start a new party system, and could be called the founder of the Democratic party.
He ran — and lost — as a Democratic-Republican candidate in 1824, out of a field of four serious candidates, all belonging to the Democratic-Republican party (since the Federalist party had disintegrated by that point), including Adams II.
The aftermath of that election saw the Democratic-Republican party split into factions around Adams (the winner) and Jackson (the person who had received the most support from the people). The Jackson faction became called the Democratic party. Adams’ supporters were briefly known as the “National Republicans,” but weren’t directly connected to the later Republican party founded in the 1850s.