I did the same thing as you, and it worked as well! Another thing I did that really showed my parents I was thinking of the 'what ifs' and things that could go wrong and how to plan for them, was made a list with things that could go wrong and how to solve them. Like one was since my barn is 45 minutes away if my horse got seriously injured how I would be able to tend to her without driving out everyday and I got permission from my friends that live 10 min away and own horses, that if it was ever needed I could keep her there and take care of her there until she healed. And I too had to get a job and put a part of my check each month to an emergency fund. Also, we don't own a trailer so I talked to horsey friends who agreed to trailer the horse to the barn for us. Just things like that really show to your parents you're responsible and thinking of every aspect of owning a horse, the good&bad. First they agreed to a lease, but only for the summer...then winter rolled around and we still had the horse, but they said you'll never own a horse until you're out on your own...and now I own my very own, beautiful mare! The financial committment is sometimes hard as you can't just spend your money on whatever you feel like, like other teenagers, but it is SO worth it! Good luck, I hope everything works out for you :)
"Don't flatter yourself cowboy..I was lookin at your horse."