Have all facts. The cost not only to buy the horse, but to stable, feed, and vet it over the next year. Talk to others who have them and ask them how much time and money is involved. Once you have all your facts than you can make a nice presentation to your parents.
The best way is probably for you to volunteer or start working at a barn that lets you take care of and train horses, or look at taking care of you friends horses or people's horses in the neighbourhood when they go on vacation, etc. You could try getting good grades, etc but it's not as convincing. If you live in the country you could as someone if you could care for their horses for a little money or for free to get some more experience. Once you have more experience with horses, they will let you get one. :) Good luck!
Horses are cheap. It's the feed & stable fees that are expensive. Unless your backyard is 20 acres & has a barn / stable (& animal rights!), I suggest you wait 'til your adult years to own & care for a horse.
Get a job to pay for the exhorbitant costs involved in owning and maintaining a horse, as well as to prove you are capable of the work required to take care of a horse. They require lots of time, money and work.
Do you know anything about horse care? Will you board or keep it at home? Do you know boarding costs or cost of hay/feed in your area? Average farrier cost multiplied by 6-8 weeks? Annual vet visit? Money set aside for medical emergency? (Horses ALWAYS have emergencies). Have you planned how to spend horse time/school time/homework time/chore time? The more you think things through and have answers, the better.