DISCLAIMER: My desperation and my convincing claims are what got me my cat. By no means will ANY of these things actually work for you, it really depends on how convincing you are and if your parents are actually willing to listen. I mean, if you follow my first option, there's NO WAY your parents will deny you a cat, I'm just saying.
Take these four options and go get yourself a cat!
I mean, let's be real. If you follow three out of the four options I just gave you, you're almost guaranteed a cat. The only thing you can't do is make promises you know you can't keep. Don't convince your parents you'll take care of the cat just to have your parents do all the work when you get it. You want the cat, you take responsibility. But I'll warn you, although cats might be easier than dogs, they can still drive you up the wall with annoyance. Prepare yourself.
Just know I warned you.