How to store parsley?
The way my parents always did it is to cut it up and then freeze it. Whenever you need some just take the box out, crumble a bit of the frozen parsley into whatever you need it for, then put the box back in the freezer.

Obviously it won't be fresh, so I suggest you only cut the amount that you don't think you'll be able to use before it goes bad, and keep the rest in the fridge, maybe with ice water as per Tetsujin's suggestion. But it does keep basically forever while frozen.