I have a question that might give us a hint - did you buy a parsley 'plant' that actually had a whole lot of small plants in the pot? Reason I asked is that I caught my daughter with one that she got at the grocery store like that, and once I thinned out the weaker small plants and spaced out the big ones, they just jumped ahead. She's in Utah, so similar hot temperatures to you (but you are more humid of course.) Look under the foliage to see if there is one main root with a lot of stems coming out of it, or a clump of small plants. If there is, you need to thin out a bunch of them to give the healthier ones a better chance at life. (soak well, then try to pull out the weaklings) Parsley has one strong taproot, so often they will thin out fairly easily.
In future if you buy more parsley, thin them and spread the plants around right away, with at least a few inches between plants so you'll grow much heartier plants.
Also if you can turn the pot around so that the Basil is shading the parsley from the hottest sun, early to mid-afternoon, I think you'll find it will be happier.