First, let me say, I totally get where you are at. I got a rocking horse for my 1st birthday, and taught myself to climb out of my crib so I could ride it all night long by the time I was 14 months old. I have been a Horse Person since before I was old enough to say the word "horse". My parents are NOT Horse People and wouldn't even let me take riding lessons when I was a teen - not because they thought it was dangerous, but because they thought it was useless.
Being a Horse Person in a family where no one else gets it totally sucks.
You've gotten some good advice for dealing with parents, for the job, and some good explanations about the cost of the horse, etc. from the other people who have posted responses to you. And I'll say this: all that advice is good, and it's right.
However, it may be that your folks just don't *get* it about horses, and it may be (possibly) that they never *will* get it. And if that's the case, it will continue to suck for another 3 or 4 years (I think you said you were 15).
Just remember, every year that passes brings you closer to adulthood. I know that 3 or 4 years may feel like *forever* (it did to me when I was in high school) but in the big scheme of things, it is really a VERY short period of time. And if you are that responsible, and if you are that serious about having horses, in a VERY short amount of time you will be an adult and will be capable of making your own decisions and getting your own bloody horse. This is the beautiful thing about being a grown-up - you can do that kind of stuff if you want to, as long as you can afford it. Because...of course...you would not get a horse if you could not afford to feed it, keep it, and pay the vet bills, right?
So...the time will come, a lot faster than you think, where you will not have to ask your parents for this. If they aren't receptive to the strategies that other people have suggested above, I'd suggest to you that you keep taking lessons (you do that, right?) and work at a barn to get experience, save your money, and when you can, get your own boy (or mare). I do promise you, that moment will arrive so much sooner than you think it will...