It sounds as if your parsley has decided to bolt. If the weather has been hot/humid, parsley will react by quitting leaf production.
Taste the leaves--if they taste like parsley, you can harvest.
Here in Texas, the fall planted flat-leaf lasted all through the winter, then bolted this spring. The spring planted parsley is beginning to show signs of quitting since we are having temps in the 70's at night and 90's during the day.
Parsley is a cool season crop, and you may have to wait and replant in the fall. Flat leaf parsley is worth the wait--the flavor is wonderful!

You can leave the plants if you have enough room. Some of the butterflies like parsley to hatch their eggs on. If you leave it you may see small black dots that slowly get bigger. They will finally start looking like catapillers in a week or so. Depending on how many cats you have they may just trim your plant or eat it all.

If they leave some and seeds delvelop you may have many seedlings next year to transplant or give away.

There are a number of very tiny beneficial insects that need tiny flowers to feed on. You'd benefit nature if you let the flowers bloom.

It's in its second year, and therefore doing what it's supposed to do. It's a biennial, and it's going to seed before it dies. Let it happen. The seeds could take some time before they ripen, but some of them will spread far and wide, and you'll end up with more plants - eventually. Probably in the most unexpected places!

You can eat the leaves, but they'll be much coarser in flavour and more bitter than they used to be.

Right. That is what biannual means.
Now go ahead plant some seeds around them.

You can collect your own seeds or use green seed floretts in cooking. They are even tastier than parsley leaves.
The root is just as good as parsnip or celery root.

To collect the seeds, pull it up when the sprigs become yellow and the seed pods are about to dry up; Hang it upside down to dry up completely. There you have it, lots of seeds. Winter sow some in your garden. They will grow the following spring. The seeds can be used in cooking like anise/fennel seeds as well. Try chewing some, to get rid of garlic smell, freshen breath.

Sorry, Cyrus, I have to disagree with you again. 'Biannual' means it occurs twice in one year. 'Biennial' means it occurs every 2 years. A big difference! Parsley is a biennial, so it lives only 2 years, going to seed in its second, before it dies.

No seed will germinate if it's not fully ripe (mature). So leave the flowers until the umbel is completely brown and dead-looking before trying to collect the seeds from it.

If you want to use the roots of parsley, the Hamburg parsley is the one to grow. The roots are much bigger. Parsley roots taste like a cross between a carrot and a parsnip - with parsley.

And if you eat the seeds, go easy with them! Too many of them can cause liver and kidney damage, and haemorrhage. Never eat parsley (any part of the plant, but particularly roots and seeds) in any quantity, and especially not in medicinal amounts if you are pregnant - it's an abortifacient.