I think the Democrats and the British Tories are the only parties in the world right now that are older; they're definitely the only major ones that come to mind.
Strictly speaking even the Tories aren't as old.
You sure? Didn't they get their start in the late 17th century as the party most sympathetic to the monarchy, with roots in the Cavalier faction?
The Conservative Party proper was not founded until 1834. If we go the "earliest predecessor" route, the Democrats and Republicans are equally old, and the oldest political party in the world is actually the Liberal Democrats...