If your parents have said no to getting you a horse, I don't think there's much advice that we, strangers from the internet, can offer to sway them.
I know you want a horse RIGHT now. But sometimes we don't just 'get' what we want in life.
Not that it really matters, but just how young are you? Are you absolutely sure there is NO possible way for you to work and save up a bit of money? Can you cut grass, babysit, do chores for different people in the neighbourhood? Even saving up pocket change can make a big difference. Though you can't get a 'real' job yet, I bet there are tons of ways you can earn a bit of cash. And by working and trying so hard, you'll be able to prove to your parents how responsible, dedicated and serious about this you are. Not only that, you'll be able to prove to YOURSELF that this is something you really want.
Your parents may never buy you a horse, that's just the way it is. If you really want one, you'll have to do it yourself and find a way. You won't get one tomorrow, it may not even be for a few years, but when you do finally have the money for one, don't you think it'll be that much more special? The horse will be the fruits of your labour; it won't just have been a gift from your parents because you 'wanted' it.