Volunteer at places near you and always be involved with horses. Show them that you can be responsible around them and that you are improving in your riding. Always show them what you can do. That's what I did and I'm picking up my new horse Sunday!
When I asked my parents they said no but, I started riding and working more with horses and they saw how happy I was, that I wasn't so negative or depressed anymore and I was more confident around people. That was a huge deciding factor for my parents because they just want me to be happy. Also I try to pay for everything for the horses.
I have problems convinceing my parents to buy me a horse believe me its hard to convince them lol. Just tell your parents that you have been working hard and that you can take care of another horse and that you can be responsible.