Research... What kind of horse you want, cost of such horse? Best places to find tack and equipment you'll need. Show them how you've checked auctions, catalogs, Craigslist, etc to get the best deal. Where will you board? Look up places, find out costs and what's included. Check out large animal vets in your area, costs for annual exams, vaccinations. Research farriers, cost of trims/shoeing. Figure out your current schedule. School hours, homework hours, when you'll have time for a horse. Save your money! Help with the costs associated with buying in some manner to show them you are willing to put your money where your mouth is.
First, learn everything there is to know about horses. Second, show your parents in real life, ex.: Go to a farm and care for a horse all day. Then, buy a field and stable for your horse, and have another $10,000 for a horse. Good luck, LOL.