Whether coaching helps or not is completely the wrong argument to be having. That is the real nexus of the issue here.
First off, your parents are right. Coaching helps! I worked at high north for 6 years and was a camper for 3 before that. The things I learned in those years made me 10X the skier that I was before. Having someone help you out with your tricks is for sure helpful.
However, by no means do you need formal coaching. By no means does this make your experience any more fun. When you're freeskiing, you don't have to win the olympics... you don't have to win local contests... hell you don't even have to be good if you don't want to. If you want, you can just ride with your buddies and throw straight airs in the park your whole life and you have every bit as much right to call yourself a freeskier. You're free to do whatever the fuck you want... including not being coached.
Your parents likely grew up in a generation before, where there was a much deeper emphasis on competition. There wasn't the same kind of outlets we have today, such as video parts. Someone making a video part uses skiing more as artistic expression, so they are just as much artist as they are athlete. A coach can teach you how to pop properly, and how to do a better job stomping your tricks, but will fail at teaching you to have a personal style that is unique to you.
So if you want to 'win' the argument, simply bring up the fact that yes - they are in fact correct. If you wanted to win the X-games, likely a bit of coaching would help you get up to the ridiculously high level needed to win that contest. However, if you choose to progress your skiing in the direction of video parts, then coaching could hinder your own personal unique style. Perhaps it could assist on a few basic mechanics, but it is by no means the be-all-end-all.
Really in the end a lot of us end up coaching each other. I know my buddies and I always used to teach each other stuff, and pick things up from guys around us. In a way our culture helps coach each other as we go. I'll always let someone who is doing their first 360's they need to pop a bit harder off the jump if they want to stop landing backseat. I'm not coaching... rather just helping out a fellow newschooler.
Conclusion? Coaching doesn't always suck... sometimes even just shredding hard with a super rad skier who happens to be able to explain themselves really well is a really good experience.
Bottom line though, skiing is whatever the fuck you want it to be.