Some horses u can lease for a while and then buy if u enjoy them
Make a budget. List every single item, your farrier bills, vaccination bills, medicak bills, hauling bills, all of it. Then be patient. It seems like the end of the workd, but it isn't. They simply may not have the spare 5-10k per uear
Shadow vets, farriers, trainers and work at barns and show your commitment.
When I tried to convince my parents there first answer was no. And I did not take no. I here friend that had a horse for sale and I had then come out and see me with the horse and then I made a pro and cons list I had 65 pros And 4 cons. Those are just some ideas
Horses take all your time and money. No time and money for boys, partying, a social life, etc!
Show them you are seriously into this. Make sure you have taken lessons for a couple years and have helped around your barn with turnout and feeding. Make sure you take good care of the pets you already have and do chores around the house. Help your parents pay for it if you can or do extra work at the barn for reduced board. Make sure you know and equine vet and a farrier and know someone who can do floatings. Make sure you have most of the details sorted out before you get a horse so you can be prepared :)