I’m going to echo what the other posters have said but add another perspective. I wanted to be a professional, too, but discovered that it wasn’t nearly as much fun once I had to ride for a living. When I was your age, I didn’t realize how hard it is to pay bills, support myself, get food on the table, all the things my parents just took care of. I thought juggling school and horses was hard. It’s nothing like trying to figure out how to pay the electric bill because your client’s check was late, or having to ride horses that you can’t stand in terrible weather when you’re sick because they’re the only 2 paying horses in the barn. It sucks to be broke, and I really strongly suggest that you have something to fall back on (or fall into; I have no problem with putting off college and giving the horse thing a try, but make sure you have a nest egg so you CAN quit and go to school. Don’t be working at a slightly sleezier version of Hooters, like me :)).
So I will give you some ideas about how to get more rides on more horses. I was in your shoes at 14 - parents who couldn’t support the addiction, but gave me a lot of freedom to pursue my passion. In a strange way, I lucked out in the gene pool. I was an extraordinarily small teenager, and I’m so old there were no age restrictions on riding ponies - as long as you were a junior, any pony was fair game. So I had a career riding bad ponies - even smalls and mediums - until I lost my junior status (yes I was riding smalls after I graduated from high school!), because when I was 18, I was the size of your average 10 year old. If you can find a niche like that to get into, you may find people handing you rides, even if you aren’t “local.” My trainer got people to bring the ponies to us!
When I was 17, I wanted sooo badly to do medal/maclay classes, but didn’t have a horse. My trainer, who was getting me all the ponies to ride, put me in touch with an amateur who had a jumper she was kind of afraid of. Plus, she had more money than time. We made a deal: I could ride her horse in the medal and maclay classes if I would pack her stuff, clean her tack, make sure the horse was clean, warm him up and get him over the fences before she showed up at 4 pm to get on for her class. Then she would throw me the reins and go get a glass of wine 
It was a fair deal - I had more time than money, and she had more money than time!
Another good way to get something to ride – work your connections, be reliable, be honest (and mature, and poised) and have something to offer (I would say your best bets are your time and the attention to detail I strongly urge you to develop). I learned how to braid, and when I started braiding for people outside my own barn, I was able to make friends with new people who had new ponies that needed riding (since the ones I was riding were usually always for sale, and the better job I did, the faster they sold – so I had each new pony for progressively shorter periods of time!!). If you show that you are reliable, polite, ambitious and trustworthy, lots of doors will open. And trust me, the more horses you ride, the better your skills will become; you will learn that every horse is an individual, and how to get the best out of each one. That will help you get better raw material for less money as you get older; if you have the skills to deal with a horse with a hole, or a major greenie, you may be able to get into the sport as an amateur (or even as a professional, should you go that route) for much less than a nervous nancy who HAS to have a packer!
I personally would find you, assuming you are as polite, humble, earnest and dedicated in real life as you are here, to be a breath of fresh air given the entitled brats I mostly see at my local barns. I would be willing to give any kid who proves to me that she’s serious and won’t flake on me a hand up, the same way I was given a hand up! And I’m sure I can’t be the only one!
