They're black swallowtail caterpillars. They get preyed on/eaten by predators quite a lot, but I can't advise you on how to raise them yourself on parsley, never having raised any on parsley myself.

Maybe somebody else can advise you on this.

Sherry

They are the favorites of predator wasp, as previously mentioned.
Unlike some here, I let nature take her course, but since my st's are scare this year, I caved in and put screening (cut into lengths) around my remaining cat babies for protection.

These caterpillars are pretty easy to raise inside. Just make sure that you have enough parsley to feed them.

They are beautiful butterflies.

Here is a link that might be useful: Black Swallowtail

It's up to you whether you want to bring them in or not. Unlike Monarchs, Black Swallowtails can overwinter as a chrysalis. So, they might not emerge until next year. They are susceptible to predation, though. I visited my SIL and found two late-instar cats on her parsley. I brought them home, mostly because she isn't an outdoor type person and wasn't thrilled to have something eating her cooking herbs. I'll feed them until they pupate and then put them in a protected spot outside.

Martha

I brought them inside and the parsley is holding up well in the water. Thank you for all your help. I'll keep you posted.

A beautiful caterpillar photo.