Growing up, I asked for a horse for every birthday and Christmas until I moved out to go to college. My parents always said no, that it was a waste of money. The only time that I got to ride was when we would visit my grandmother about once a month.
Now, I'm grown up and have 4 kids of my own. We did get the kids a pony when they were little, but the kids outgrew the pony and three of them lost interest. So, my 2nd oldest has been taking care of the pony by himself for the past few years. He has begged us for a horse that he could ride for sometime now and said that he would pay for the horse and everything else. He saves his money (birthday money and odd job money) and it was possible for him to help out because we own our own land and wouldn't have to pay boarding every month. He has also worked with the pony, who is blind in one eye (due to a birth defect) and mostly deaf. She was a terror, but is now very sweet and very easy to work with.
My husband was against getting him a horse because he quit football and baseball because those activities weren't his "thing" and he figured that this would just be something else that he would quit. But, after seeing how hard he worked, thought that maybe horses WERE his "thing".
So, we thought that he proved himself and we bought a horse for all the kids to share. Now, a couple months later, my son is the only one who does anything with the horse. The other kids quickly lost interest when they realized that their brother did more than just throw food in a bucket for the pony. So, he is now the only one taking care of the horses and he is doing a very good job at it. He also just bought a new/used roping saddle this past weekend with his own money.
So, my advice is to be patient, work hard (save your money) and prove to your parents that you are mature enough to own a horse. Because, I tell you what, the begging didn't work with my son...it was his hard work and dedication that changed our minds about getting him his horse.