Pros and cons list, show your responsibility, negotiate different rules and prices with them, prove to them you're committed and it won't be a thing to toss aside. Put all the prices down and explain how you can reduce it or help pay for it.
Help out around the house, get a job and show how responsible you are.
I bought my first horse at 16 and I paid for him entirely myself, and I still pay for him entirely myself. It was the only way I could get a horse. It takes a lot of dedication but it is so worth it.
It's a good time to learn how to negotiate! When I was young I made a deal with my parents that - when I baby sat for the neighbours - I would save every penny. I did do that and back then we made fifty cents an hour! They said they would match me dollar for dollar (they didn't believe my commitment). It took almost a year of babysitting and doing odd jobs but I saved $125 and they matched me. I bought a horse for $200 (which was a lot back then). I babysat every weekend - when I wasn't riding thereafter. So my parents were happy that I was busy and not getting into trouble. I was sooooo happy because I got a horse!
Maybe they would be more open to leasing a horse! If all goes well you could look into buying that horse (if possible) or start trying other horses :)
Show them how committed and responsible you are
It's important to consider the costs of owning a horse and the costs it will result in your parents paying. It is extremely expensive. The horse itself is usually cheap - it's the tack, stable rent, feed and numerous random things you end up buying for your horse that is expensive! Furthermore, it's hard work. You have to be there every day, working with your horse, brushing, cleaning out stables, poo-picking fields, riding, vet vaccinations, farrier work, etc. Once you buy a horse, it's a life commitment. This is all very off-putting for parents and most aren't prepared to take on the role of becoming a responsible horse owner.
If money and time is not a problem, and neither is age (if you're still in school/college it is fairly difficult to find the time to juggle looking after your horse and work) then perhaps you could convince them by volunteering at a yard, or perhaps loaning a horse to prove to them you are perfectly capable of owning your own horse.
Hope this helps!
Show the your committed show, them your responsable, by doing chores around the house and maybe at the barn.
show commitment to your riding and maybe find a stable and help out with the chores. at the stable i go to they do own a pony once a year and you get to take care of a horse for the whole day
It takes a while. Try putting together a slide show. Pros and cons, what you would do to help out/keep up your part. Good luck!!