You have to prove that you are VERY VERY VERY responsible! Having your own horse is alot more work than using a lesson horse, or leasing a horse, and tonnes more expensive. Sit down and talk to your parents about the pros and cons about getting your own horse, and say that it would be a good expirience for you to have, and that you are getting more serious into riding. I'd say, don't convince them, let them convince themselves, because when someone else does the convincing, it usually ends up as a regret. Take your time too. This is a huge decision, and if they say yes, take your time in choosing a horse. You want the right kind of horse that suits you.
Good luck!

Talk to your parents and trainer about a lease or half lease. It would be a good way to show your parents that you are serious and could take care of the horse without jumping all in. Think about what your parents will ask you when you approach them and try to have answers ready. That will also show you are serious. Good Luck! :)

I had to show that I could take care and provide for my horse. I also had to show that I was very responsible and made sure I wasn't failing in school. So when they saw I can they bought me a horse.

I had to work very hard to convince my parents! I had to have almost perfect grades and show I was responsible enough to take care of a horse!

Lol, I'm a mother and my son wants a pony.
The first question you must ask is, can your parents afford the upkeep of a horse? Many people aren't aware of the exorbitant cost of keeping a horse - especially when it comes to vet fees.
The second question is, who will look after the horse? It's all well and good buying a horse, but they also require a lot of time and effort. Too many people get pets from their parents only for the parents to be left with all the looking after.
If this all seems very depressing, let me reassure you, it won't be long before you will be an adult with your own income (it might feel like centuries away now!!) and then you can buy as many horses as you like.
All I will say is, don't put too much pressure on your parents, I'm sure they would buy you a horse of your own if circumstance allowed them to. Parents really don't like not giving their kids the things they want, but sometimes, they don't have a choice in the matter.