Well, so... the parents are very competitive and willing to put dollars into a new horse, but not willing to keep the old one as well. What I see is that your parents want you to focus exclusively on the horse you will be using in competition. And this will be horse number what exactly?
As mentioned there will ALWAYS be something wrong with any horse. Are your parents just going to keep trading them in? That is not the way to excel at competitions. I can't imagine the number of hours that top notch competitors have on each of their horses. The horse industry is not: keep a horse for a year or two, then ditch it for a newer model.
Yes, I can understand not having resources closer to you. That is my world as well. But you have a coach -- your coach should instruct you on how to handle your horse's bucking issues. Your coach can't teach your independently of your horse. That makes no sense. When the next horse arrives and that one doesn't like pink flags at the shows and throws a hissy fit at every show, if your coach won't help you now, she certainly won't help you then either.
Do you want a new horse too? If so, is it ONLY because of the bucking? If your horse, other than the bucking at new places, is working well, IMO, it is cheaper and money better spent to send the horse out to a trainer for even a month, let you go out there for a week or so afterwards and get some info/tips from the trainer. But if you want a new horse too, well, then me thinks you'll just have to suck it up and say bye-bye to lady bucker.
I think it is reasonable for your parents to limit you to one horse.