Sorry OP, I get that your circumstances aren’t ideal right now but this school is not the golden ticket you’re making it out to be. Boarding school and an equine science degree aren’t how people break into the equine industry and won’t carry much weight with potential employers. You’d be much better off doing as much as you can with your current horse while finishing high school and then taking a gap year or two as a working student to see if the career path is for you. If it turns out not to be, you’ll want a more relevant degree on hand for a career to support horses as an amateur.
The level of riding and showing you’re talking about just isn’t feasible for most families, it isn’t just you. The fact that you own a horse and made it to 3’3 already puts you ahead of so many people. Can you find an instructor that will come to your farm or do virtual lessons with you just to start building out a network? You’d be surprised what opportunities come available through word of mouth. No one is suggesting your parents uproot their lives for you, but you may find ways to spend a few weeks here and there immersed in a serious program. If nothing else it gives you a head start on a working student position later.
In any case, if your parents were committed to the idea of this school they would be figuring out the financing themselves, not leaving it up to you. This could be a good chance to have an adult conversation with them about the family finances, just to get some more realistic insights into what it would mean to go for this. I spent a lot of time as a kid being annoyed at my parents for not spending more on my riding goals but that was because I had no concept of what those numbers really meant. Looking back now I can see how unrealistic I was, mostly because I had absolutely no frame of reference to work from. I saw other kids buying horses and going to shows so I assumed we could do it too, which just wasn’t true. Even if we could have it wouldn’t have been fair to take up that much money just for myself without thinking of my parents and siblings. Even now I sometimes can’t believe how much I spend on my horse, even though I can totally afford it. It’s just an absurdly expensive sport, and if you want to make it a lifelong endeavor you’ll have to come to terms with having financial limits and finding a way to enjoy the experience in spite of them.