As far as convincing parents, I asked about 20 times a day for 3 years and they eventually gave in. I had to sign a document stating I would go out there everyday to care for my horse and keep at least a B average at school. Even then they weren't prepared for the responsibility that THEY had to take on.
If you're old enough to get a job - then get one and save, save, save. Show them your savings record and show them how seriously you are taking this. Find your way out to the horse everyday, commit to the things your parents find important too - improve your school work, spend time with family etc. Give them no reason to doubt you.