Try leasing a horse first. Once they see you are responsible enough on your lease days and that you have enough experience to own your own you should win them over! Leasing is always fantastic option though!
Get ur grades up, and do extra work like everyone else said to prove to them!
Show them that you're responsible and that you know what it takes to have one. Plus I feel like your parents know when you're ready to own a horse.
show them you are responsible, have all the expenses figured out, and that you have a job to support the horse
Show them you are responsible and that you'll help pay for the horse. Not for the entire horse but yeah.