In your zone, right now you can directly sow them in the garden. But parsleys like cooler weather. I consider them cool crop. I started mine in cold fram in February. I am in zone 7/8 GA. I have had one harvest so far. I chpped and freezed them. I am practical about herbs. I plant them for consumption not just for show. I have a parsley-like herb called "chervil". I found the seeds by accident in a dollar store. It is a french favorite in place of parsley. It has tiny fern-like leaves and grows fast and taller than parsley.
If you like parsley you should also plant some chervil.

My parsley is going to seed. It's definitely a cool season herb down here.

Starting parsley indoors vs. outdoors...I always start my parsley in situ outdoors. It produces later than indoor starts but I always have good luck with it outdoors. The one caveat is that about the time you are ready to replant that area with something else because you are sure this was a failure is when the parsley comes up.

Patience, Little Grasshopper, the parsley will come up.

FataMorgana