Short answer
Parsley is safe for dogs, so training isn't strictly necessary for health reasons. However, teaching a solid 'leave it' command is excellent general practice for any plant in your yard. If your dog does eat some, just offer fresh water and watch for any unusual stomach upset. Call your own vet if anything seems off.
Parsley — Food safety facts
Safe to share · Dogs · full breakdown on the linked food page
Safe, and a little is even used as a breath freshener for dogs.
The full verdict, portion guidance and serving notes are collected on the Parsley food facts page.