You have two problems that are at play here, actually aggravating each other.
First, etiolation - lack of light and the side effects. We discussed this before in your other question.
Second, those seedlings are too many, too close together. This means that they will complete for the resources like water, light and especially space. A factor that puts stress on them and which will ultimately kill many of them.
And both factors are creating a vicious cycle: The closeness encourages etiolation (seedlings trying to get “over” the others to get light), etiolated plants are weaker in comparison.
So what now?
Tackle problem 2 first:
You need to drastically thin your seedlings. You may try to take a lump out of the tray, gently separate and pot up individually or with a lot more distance. Unfortunately, the seedlings look very fragile and might not deal with that procedure too well - and if they do, you have a lot of parsley at your hands. I personally would just pull or cut about two thirds or even more of the seedlings.
Then address problem 1:
Now that you have removed the competition, give the remaining plants a fair chance. Supply enough light, ideally outside. But don’t put them straight into full sunlight or they are literally toast. Grow lights are another option.
And finally, the watering part: Never let the seedlings get “wet feet”. Make sure that all excess water can drain freely. That’s why good pots have holes at the bottom. This plus removing the dead material should keep the fungus in check as well.