Unfortunately, you can't change your parents. However I do think they have a point. I began riding when I was 5. I took lessons once a week until I was 8. When I was 8, I began leasing the bratty lesson pony I rode for a 3 day/week lease. He taught me a TON about riding in the year I leased him. When that trainer decided to switch barns, we looked into purchasing a horse. I got my first horse when I was 9. He was (at the time) a 10 year old Paint gelding, who we still have. I owned him for about a year, and then when I was 10 I began leasing another very young pony mare. (My sister also rides, so she started riding our paint) She was a green broke 4 year old who I taught to canter and jump. She also loved to buck. So I took MANY falls off of her in the summer I leased her. However, she made me quite a nice little rider. When I was 11, we bought a 4 year old Haflinger mare as a project pony for me to train. Long story short, she ended up being crazy, and knocked me out. We sold her about a month after buying her. When I was 11, I got my current mare. She was quite a handful. She was 5 when I bought her, but she acted like a 2 year old. (She's now almost 8, and still hasn't quite quieted down)
My point is, that I rode for quite a long time before I got a horse. And I leased first. I'd love to have the option to ride different horses now, because I can only ride my horses. Horses are extremely expensive. It's not just the initial cost of buying the horse. That's the cheapest part. Then you have to tack on board, vet and farrier bills, equipment, lessons, etc. You don't realize how much owning a horse costs until you actually have one. You also need to check up on your horse AT LEAST 3 days a week. If the farrier comes on a day that it's freezing cold and you're sick, you have to suck it up and go stand there with them. Owning a horse isn't always glamorous or fun. There's plenty of tasks you have to do that you'd rather not do. If your horse gets hurt or sick, you have to get the vet to come check it out. An injury may need to be bandaged or medicated every day, and gets who has to go out and take care of it? Yes, you. Whether you think you are ready or not, if your parents don't want to drive you there as often as you need to be, and they don't want to pay, you simply cannot get a horse. However, leasing may be a better option for you. It's a lot less responsibility, and it teaches you a ton. Often times you only have to pay board/leasing fee and don't have to worry about the vet and farrier bills.
This probably wasn't what you wanted to hear, but you need to be realistic. Try asking your parents why they don't want you to lease a horse. If its a money issue, maybe you can offer to do extra chores, or earn money outside your house (baby sitting, pet sitting, yard work, etc.) and offer to pay for part ofhte costs every month. Good luck :)