lol, convincing people to do things is practicly my CAREER. it's easy for me.
Probably like Gracie said, volenteer, maybe get books on horses, and study them (and leave a few plainly but not TO obviously around the house) this lets your parents find out for themselves (saving face) that your studying, then be SUPER nice, once again, not TO obviously, help a little bit more around the house, occasionally (again don't push it or your parents will get suspicious) maybe buy them a present or 2 if you can. this puts them in your favor. Next (give it a month or 2) next, write a detailed (mine was 20 pages) report on horses, hide it in your room.
Then ask to do extra chores and stuff for money, wash the car, ask other people, save up as much money as you can, do NOT spend this money, save it.
give mom and dad the report.
it's also nice to make a few promises and keep them before this to:
bring these into ploy, work your heart out on school and get the best grade possible, a day or 2 after you come home with it, write a really nice, well grammared, well spelled, polite letter, telling them what you want (it's always nice to offer to help pay for expenses, and to do extra chores at home, and a few other miscelanious things. and KEEP them. Tell them that you even saved some money to help out with it.
If it doesn't work, stay low for a few days and repeat process till it does work.
I had to do this and it took me 2 years just to get a hamster, but it worked.
so the whole thing is buttering them up
(do be sure that they can afford it. The horse may be able to eat grass but you can't!