@TheLemurLady None of us here know your parents (we don’t know you either for that matter). As such, its tough to know what would sway them. More importantly, its tough to know the real reasons behind their views. May well be the case that the reasons they give you are only part of the basis and that they will not tell you other reasons (they may not even know those reasons themselves or may be unable to explain them to you). So you may be trying to reason them through an incomplete list of issues which makes the end goal likely unreachable.
Some things to consider:
You say you aren’t worried about money. How much do you know about your parents’ financial situation? Most kids in my experience don’t know a lot about financials of their parents. Some know some info but not all. May not be a clear cut as you make it sound in terms of affording boarding school.
Being comfortable with a month away at camp is very much different than being away at boarding school for 4 years. and 7 hour drive is better than a 14 hour drive. But its still a long drive. And weekend visits may well mean seeing you for a day or a day and 1/2. Long time in the car for that length visit.
Your parents may be testing your commitment. No offense, but kids your age change your mind a lot. And that is totally nature (I change my mind too and I am in my 40s). Still trying to figure out who you are and where you are going. Along the way, you likely will change directions many times. All good. But I noticed you were posting about 2 different schools here about a month ago. Now a third. And a different public school was on your list according to your OP here. Again, nothing wrong with that. But before they commit to 4 years of boarding school, your parents may want comfort that you are committed to it as well rather than just one possible direction on a list of changing possible directions in a kid’s path of life.
May be emotional issues involved here. Your parents had it in their minds that you would be leaving home in 5 years (I think you are going into 8th grade, right?). As a parent I can tell you that time flies by way too fast and there are times when you just wish that life came with a brake to slow things down a little. But now you say you want to accelerate that train even faster and leave in just a year. For many parents that would be very difficult to accept (for some time to process/accept may work; others may not get there emotionally).
You seem like a very bright and driven young person. World needs more people like that (at any age). But its even less common at your age. Follow your dreams and your passions.
But keep in mind, there are multiple different paths to success in life. Particularly for people with drive and passion. So no matter what path you take for your secondary education, I expect you will do very well.
All the best luck to you.