Other representative republics?

Excluding the U.S., are there any other non-parliamentarian representative republics?

Many. Most nations on the Americas have presidential constitutions that were drafted with the US one as rolemodel.

Just to mention a few presidential representative republics:
Mexico, Argentina, Brazil, the Philippines, in some way also France and Russia, although those are something of a mixture with a powerful President but a Prime Minister who needs parliamentary support.
Remember that “presidential” means the head of government is elected by the people directly, not by parliament. Virtually all modern democracies are representative ones, although many of them have more or less direct democracy involved. Pure direct democracies where everything is voted upon by the citizens directly is hard to do in complex states with large populations.

Cool, thanks Schnitte.

So the proper term is presidential representative republics?

Well, I just assumed you meant “presidential” when you said “non-parliamentary.”

Let me elaborate on the difference a bit. The US have a presidential system: Head of government is the President, elected by the people. The President is ruling the country and appoints his ministers (secretaries), but his power is restricted by some Parliament that’s elected by the people just as well.
A parliamentary system might have a directly elected President as head of state, too, but usually this President has very limited powers - in some countries only ceremonial duties, eg shaking hands, giving speeches and formally signing laws, documents and treaties decided upon by someone else, without having any veto rights. The real power is exercised by some Prime Minister (or Chancellor, or whatever you want to call him) who is elected by Parliament, which in turn is elected by the people. The nice thing here is that Parliament can fire the government whenever it wants to, unlike in the US where the President doesn’t need to have a parliamentary majority - he cannot be fired (unless of course he breaks the law, in which case he’ll be impeached, but a Prime Minister in a parliamentary system will lose his job when he simply loses majority in Parliament). OTOH, Parliament will get dissolved if absolutely no stable majorities can be formed to carry a government.

In Europe, most countries have a parliamentary system. When the former Spanish colonies in South America gained independence in the 1820s, many of them decided to establish a political system similar to the US one, which is why there are so many presidential republics in Latin America. Former British colonies, such as Canada, often have parliamentary systems because Britian is the parliamentary state par excellence. I think globally speaking, the parliamentary system is a bit more common than the presidential one. There are mixed systems too.