Have you inspected for caterpillars?

We sure didn't see any, and they should be easy to spot now that most of the branches are bare. We'll look again in the morning.
We've been growing parsley for years, and have never seen so much damage (unless there were slugs or snails).
I told my husband that perhaps the nursery put a few slug eggs in the middle of each plant (where we didn't throw bait) so that we'd have to go back and buy more!

Checking the parsley this morning, I spotted 1 dead slug and no caterpillers as far as I could see.
Since there are 18 half/eaten plants, it seems that just a couple of slugs/snails couldn't do all that damage, and the dead slug was not unusually fat or large, either. Are some slugs now immune to bait???

Beer traps work very well versus slugs and snails. Rabbits and other four-leggeds could be helping your parsley plants as well.

FataMorgana

A good deterrent for snails and slugs are crushed egg shells, or anything sharp and jagged really. They're soft bodied creatures, and they don't particularly like sliding over broken eggshells any more than we like walking barefoot on broken glass. Nice ring a couple inches wide around your plants should help keep them away.

Edymnion, that sounds like a great idea; I'll try it.
BTW, I've seen no new damage the last couple of days, and no new dead bodies either.
Still, it seems like the damage to 18 plants was a lot for 1 snail and 1 slug to do in one night, but that's all I see.