Are these worms on the plants or in the dirt? What do they look like?

Petroselinum crispum is a host plant for some swallowtail butterfly larva. Their caterpillars are black and green striped with yellow dots. Host plant means that the caterpillars eat that particular plant before making a coccoon and turning into a butterfly. The caterpillars of different butterflies (and moths) use different plants as hosts, depending on which kind they are.

I linked in a blog post that shows black swallowtail caterpillars in a parsley patch. The caterpillars are distinctive and easy to spot. With the caterpillars, the foliage disappears but doesn't wilt.

I must admit to not being up on all pests that could be attacking the roots. I know there are destructive nematodes (as well as beneficial ones) but I believe they are generally below what can be seen by the human eye.

FataMorgana

Here is a link that might be useful: Recently seen in the parsley patch...

Ummm .. they are worms, not caterpillars.
Yes they are attacking the roots. They look like nematodes, but they are large enough to be visible (but too small to be photographed properly)

Only two more parsley plants left, and both are under attack. No italian food this season :(

could it be carrot root fly larvae? Not sure if it goes after parsley - but they are the same family.