I'm working and getting a $28k bonus for evading the gift tax--that's fine for now. My parents would know about that part. I have the pressure to present any business ideas to my parents, or even the trustee. Video games frankly seem tempting.
IMO the smart thing to do would be to be cautious about choosing your industry (that's assuming you want to be entrepreneur, which may change with time). Video games development is definitely not for the faint of heart (i.e. when you're starting out, to turn a profit you need to be either very very good AND lucky or just plain super-lucky), so I'd suggest spending some time in the industry as an employee first to assess whether you like what you're getting into. This has the benefit of being exposed to a lot of things, so, in the end, even if you come to conclusion that you don't want to start a game dev company, you might figure that you really liked what programmers|artists|musicians|managers|marketers did in that company and choose it as your next pursuit. Life is a journey really and it's best to treat it as such from the start.
Once they're reduced to nuts and bolts and hard work, they become much less interesting.
If that doesn't interest you, you could start a company or go to school. You don't have the same pressure to make money that many recent grads do, because you have a big safety net if you fail. So the risk/reward ratio is higher for you in startup land.
I don't see why rich parents should change your behavior. Don't you want to be successful on your own? Or do you just want to wait for an inheritance (read: wait for both of your parents to die)?
What would your parents think if they read this post?