Be aware of your families budget and income, and be realistic.
Know your costs to buy v. Lease v. Taking lessons.
What I did to convince my parents was I got a smaller pet to start with. I got a rabbit and when my parents saw how well I could take care of it they let me get a horse!
Offer to pay for part of your horses care when convinceing them!!! I now how you feel I would get a horse but I just can't afford it right now! And my parents can't help me!
Either have your parents pay for the major stuff (i.e. Horse, trailer, vet) and you pay for board and feed and stuff like that or help them pay for everything. Make sure they know you are entirely dedicated to this. Some parents worry about the costs. Others don't want to have to deal with it if you decide to by want it. I've been trying to get one for a while Bc I only ride once a week during lessons and I feel I need to be able to ride on my own to figure some things out. Make sure they know that owning a horse is a learning experience, you won't know everything about horse care when you buy one because you haven't been in those situations before.
Write down the pros and cons of you getting a horse.