Of course they do. Its name is "Congress". The Lower House of Parliament is called "House of Representatives" and the Upper House is called "Senate".
No, actually we don't have a parliament. BRTD's point is dead-on: if Congressmen in the U.S. were simply appointed by the party we would have way more women and minorities in office.
BRTD's point about party control is correct (although MP's are not simply "appointed" by their party, it's a bit more complicated than
that). But that's got nothing to do with the name of the US' parliament. The British parliament is fptp too (although parties have more control on who gets nominated).
Pakistan uses first past the post too btw. And that Pakistani parliament is full of Islamists.