Well you should first show them how good you are riding and doing in hand work. Then show them you are responsible and clean out the stalls, clean the tack, feed, and do other chores well. Also, leasing a horse or two (or more!) before you buy is the best way to start. Leasing different horses (at different times, of course lol) will keep you open to new personalities so when the opportunity comes up to ride someone else's horse or you need to help or ride a different one than that one you're used to, you'll know how to handle situations. Another thing is time and money. You have to be at the barn (be an experienced rider as well, which, I'm sure you are lol) all the time and be responsible for your supplies, tack and continuous feed and chore schedule as well as the pasture time and correct riding times. Also money. You should try saving for your horse unless your parents are open to buying one for you. But also boarding fees can be an issue so unless you live on land, you'll need to board your horse somewhere. Also you're riding supplies (barrels, jumps, etc.) should be taken care of (bought lol) long but suggestions :)
first & foremost, don't even bother asking if you know that your family is not in the fianancial situation to do so. many parents want to buy their kids horses, but simply can't afford it. you need to understand your financial stance before you start convincing. if you've talked through the money part of horses, then there is a point of convincing them. i recommend showing your responsibility & dedication to the sport.
To convince my parents to buy me a horse I showed them that I was able to care for all of our dogs, fish, lizard and flying squirrel. That showed that even tho they all require different needs I can meet them. So if you have some pets start taking care of everything for them.
I also told them that I would get a job as soon as I could(we got my horse 3 months before my 16TH birthday) so I could help pay for vet/farrier bills, my lessons and other items.
Also another convincing factor was the fact that I was going to be 16 in 3 months so I would be able to go to the barn by myself, and my parents wouldn't have to take the time to drive me there.
I'm also very serious about riding(tho I know how to have fun) and my parents knew that I wanted to progress in riding and the school horses I were riding were not allowing me to progress anymore and my parents wanted to see me be able to learn more so yea that's kind how I got my boy! Hope this gives you some ideas! Good luck!!
Show them how responsible you are and do chores without them asking. Or sit down and have a deep conversation and tell them about the horse