Parsley is a biennal. Puts up leaves first year, puts up seed stalks second year then dies. Plant new plants while waiting for the seeds to become ripe.

Short answer - yes. If you leave it alone it might self seed if it can pentrate that mulch. Meanwhile start or buy some new plants.

Parsley will go to seed early if it has been subjected to late transplanting, or a sudden change in temperature, or other 'abuse'. Also (something to watch for) some unscrupulous nurseries will sell you small seedlings that are actually more than a year old and treated with anti-growth hormones. So when you plant them, and you think you've got a true baby, actually you've got a year-old plant instead!

Yes, your parsley is going to seed, which is a pity because the plant is very small for an adult. Let it go to seed and you could find parsley plants all over the place next year.

I agree with all said above. Only thing to add is that the tiny flowers on the umbel will attrack some useful tiny, non-stinging, bees and wasps to your garden.