Yes RSI is wayy more common with gamers and atheletes, cubing doesn't really require much physical exertion, I say that but I can't solve a Megaminx 5 times without getting tiredCubing is a cool skill, and I've even heard of people getting jobs in part because the interviewer was impressed by their solving ability. But overall, it's not a "life skill" or something that will kickstart a career—but it doesn't need to be. The point of hobbies is to be an enjoyable pastime, they don't have to be particularly useful to be fulfilling and worthwhile. Do your parents expect you to spend all your free time building businesses, looking into careers, or otherwise earning money? Your profile says you're 12 years old, so your parents having the expectation that you shouldn't spend your free time having fun—as a kid should do—is ridiculous. As long as you aren't cubing so much that your grades and/or social life suffer as a result, and you use your own money on cubes, there's no harm.
You don't need to have good reasons to cube. It's fun, and it's a hell of a lot better than some other things you could be spending your time doing—like selling drugs or getting involved in gangs—and I think your parents should be proud that your #1 hobby is something that expands your brain and keeps you away from the screen. I doubt anything I've said here will convince them, but I hope that at least you understand that they're being absurd.
(As for the RSI point, that isn't something that commonly happens to cubers, even those who solve much more frequently than you. It happens to some people, but as long as you take breaks when your hands get sore and don't push yourself, you'll be fine.)