I’m going to echo what most others have said here - I know it seems like we are a bunch of old meanies who do not want to help, but we have the advantage of experience and hindsight that is just not possible for most high school age teenagers.
It is just so very, very unlikely that you would be able to put away anywhere near the $50k that your parents have outlined as the number you need to contribute. You’d never be able to ride or do ANYTHING else for the next 5 months and you’d still fall horribly and depressingly short of your goal. Let me put it again in the terms others have - to make $20k in 5 months you’d have to make about $175 (to cover the taxes you’ll have to pay) a day EVERY SINGLE DAY for the next 5 months without a SINGLE day off. That’s $17.50 an hour for 10 hours a day. EVERY SINGLE DAY. More than double that amount for the $50k. And spent ZERO of that money on anything fun, including your horse or showing.
Also - the school saying they support any showing you might want to do - I’ll wager quite a bit of money that you’re footing the bill for any of this kind of showing. Do you have a reasonable grasp on just how much money it takes to qualify for finals? You’re very likely paying for entries, nomination fees, stabling, hotels, travel for any coaches coming with you, transportation for the horse, etc, etc, etc, When I was showing at a higher level and further afield than any of my barnmates - I had my own truck and trailer, and was self-sufficient enough to go without trainer support. For championships, I contracted with a clinician that I regularly rode with for one-off coaching at the show that she was already at with a big group of clients to cut down on my expenses. I would have had to foot the entire bill for my local trainer (lodging, food, day fee, etc…) if I brought her with me. So even though you want to show, could you realistically afford it?
You’ve said your parents are unwilling to let you even look for a working student position for even a short period of time. Why? I know several kids that absolutely excelled as a working student, taking their high school classes through a virtual school. Why are they willing to let you go to a boarding high school but not go work with a trainer. That alone tells me that they know you cannot possibly make the necessary money, but don’t want to be the bad guys who flat out tell you no. There are horror stories from WS arrangements, of course, but also many excellent programs. Same goes for elite prep schools. And you easily may find you hate a boarding school as much as your rural TX school, just for different reasons. You’re going to be very poor compared to most of your classmates and that gets old and potentially isolating very quickly.
You’d be much further ahead focusing on your mare, seeing if you can work at some of the local barns - even though they are not H/J barns - as hands on experience is never a bad thing, even if you learn how NOT to do things.
Academically, really look into supplementing your local school with either community college or online classes. Prove to your parents how hard you are willing to work. Graduate early and then get the WS/groom position that will get you much further to your goal to be a pro. Frankly, if you’re even a freshman, let alone a sophomore and have not yet had Alegbra 1? You’re behind on academics and probably would be overwhelmed by the workload and difficulty of the academics of a upper end boarding school. Those schools live and die by their Ivy League admission rate and your desire to do nothing but ride will fall to the side of making sure you’re keeping up academically.
That all being said - what does being a pro look like to you? What work would be your dream? It’s not all just big sticks Saturday night classes under the lights. That is actually a tiny percentage of H/J pros. I have a friend (in the dressage world) that did something similar to this - finished high school, did a couple years of college and then took a WS position in Ocala. She’s quite successful 10-15 years later (likely going to be short-listed for Paris), but she is INSANELY talented, and will be the first to tell you she also got VERY lucky. And has parents that could afford to support her chosen path until she could support herself.