In my honest opinion, there is something wrong on a basic level about the way these parents are acting.
Their "you must win" obsession smells of trouble and perhaps a history of unluckiness with sports of their own, or parents with the same mentality, etc.
Now, I don't necessarily think that you should move up, OP, I haven't seen you ride and therefore can't give an accurate opinion on whether or not you would be okay to move up.
Now, my father pays for the majority of my riding expenses as I have no job and no car (too young for both) and he frequently tells me not to rush things, etc. He gives me opinions on my ride, my horse, etc, but they are nearly always respectful and understanding.
I am forever grateful that he pays for me to be able to participate in this sport. I understand that without his money, I wouldn't be on a horse at all.
However, I take his opinions with a grain of salt, simply because he doesn't have experience with horses like myself or my instructor, or even my friends who ride. He doesn't understand how challenging and complicated riding is, just like most people.
Whenever I show or plan to, he always says, "It's a lot of money. I want you to have fun. Don't worry about winning, that's not important."
If I do win, he (and the rest of my family, who also support my riding in different ways) is proud and glad for me. If I don't, well, no big deal, at least I had fun.
Parents so concerned about winning are not in the right frame of mind, and in my opinion it is bad parenting and teaching bad lessons, that it's all about the ribbons, etc.
So, although I respect the sacrifices made for my riding, if my father told me I had to win/be the best, I would ask him if he would like to get on the horse and see how he does in the class/ride/etc.