Thank You for all the replies. Kendra, you do bring up a good point. (Even if it was extreme, I know that that kind of thing is so definitely possible, and can happen at random times). Please keep the perspectives coming!
So, have you brought it up yet? You have a lot of good advice already. I REALLY like alexS's adv e about asking your parents what their plan is with you and your horse life. I don't know how much they know about horses.. But I'm sure they've considered all angles and made an informed decision about what's best for you and their budget. If you honestly see it as more cost efficient to own that would be a great angle to present.
How much is board at your barn? That plus your weekly lessons will be your BASE cost in owning. Remember other monthly fees like farrier. Really.. Riding 4x a week total for $500 a month isn't too bad... Unless you're absolutely sure you'd be riding more days when you own. Life gets busy.. Sometimes the best you can do is 4x a week!
If your parents decide to just have you keep leasing, I wouldn't complain. I'm not sure how old you are but you seem to be at a stage of life where you're not very settled yet. If under 16.. Remember you'll need a car and probably a job (unless your parents pay for all that in which case they might have to reduce your horse costs to afford that). Remember you'll go off to college and have more responsibilities and may not have the time/money for a horse. It could end up being a thorn in your side to actually own and have that responsibility!
When I was younger my parents wouldn't even pay for lessons. I had to turn 16, get a car, get a job, and pay for everything myself if I wanted to ride. I couldn't afford a lease much less a horse. When I first moved out (I didn't go to college) I bought a horse. Little did I know although I was making lots of money.. The cost of living, moving around, buying cars, just generally being unstable really made it impossible for me to ride and afford to ride. Anyway, sometimes you just have to wait for what you want and make sacrifices. If you really love riding.. You'll keep at it even if it means "only" a lease.
Ok, well here's another perspective on the situation: Parents out there, (from what you can grasp of my situation), what would it take for you to be convinced / say yes, and what arguments would you have?