No, parsley can't survive the heat here, so I never plant it. Maybe somebody else will know.

Sherry

Could be either heat stress or some kind of mite. Try growing Fennel; you'll be guaranteed excellent results.
Fennel is easy to grow and provides plenty of food for growing, hungry cats.
It is also tasty in cooking:-)

It could have been the heat, since it's been cooler for the last couple of days and I think some of the leaves aren't looking quite so curled today (of course it could just be my imagination). I've read that parsley doesn't do well in the heat, although I've had it many recent summers and it's usually done all right (I've only had one die off entirely). Of course, if this one is having trouble with the warmish weather we've had lately, it may have a hard time in the summer, which is when I usually get eggs. Adding fennel is probably a good idea, although I don't know a lot about it...I'm guessing it handles the heat better? Can it be grown in pots? If so, I may wind up getting some of that too.

Fennel is easy to grow, does well in large pots, though it needs more water, and lasts a couple of years. Likes full sun, but will tolerate light shade.
You should get a producing batch by Summer for quite a few caterpillars.
I start mine from seed right about now and add to them c 4" pots later in the Spring.

I'd be looking at Mock Bishop's Weed (Ptilimnium capillaceum) since it can take the heat.