Naw, they used to be one party, the Democratic-Republican party. They grew out of the Anti-Federalist Party and outlasted the Federalists. In the 1824 election, they were the only party in the US. They ended up fracturing in two, creating the Democratic party and the Whig/Republican party.

The full severing of the political parties would occur just prior to the Civil War where Whigs splintered over slavery while Democrats supported it. Anti-slavery Whigs and Republicans consolidated together as the Whig factions further separated from each other.

So the "Grand Old Party" is a splinter faction of a splinter faction that can trace its roots back to the original parties of post-Revolution America. Republicans and Democrats are technically the same age.