This fella showed up at my parent's house in New York. I assume an escaped pet of some kind.

Yes, a Budgerigar.

If he lets you approach, you might like to try approaching him slowly. If you move an outstretched finger in front of his feet, and say “step up” in a cheery voice (a bit like you might ask a friendly dog to shake), he might step onto your finger. You can then gently put your thumb on one of his feet and move him inside. That way he can’t fly off if you want to find the owner.

My wife and I had one land on our patio a few years ago. White and turquoise. She was very pretty, but didn't care to be handled much. Posted on craigslist and local Facebook groups to find owner, but no one claimed her. She lived for about 3 years, then suddenly took I'll and died.