This all depends entirely on you, but answer a few questions and I can maybe give you my thoughts.
First off, the #1 question is do you want to do what your parents do? Don't just say yes because you're pressured into going in to it. I believe everyone should be doing what THEY want to do for the rest of their lives.
I guess the second question would be are you good/have real experience in business? I mean I'm not trying to sound rude, and I can tell by your posts that you're definitely not blind to the fact that you come from a pretty wealthy family, so I'm just wondering if they bought all that stuff for you or if you worked to get it?
To answer your question, education is most definitely needed, FOR SOME. Some people are born entrepreneurs (different than being born from entrepreneurs, remember) such as whichever one of your parents is the initial owner of the business (No offense to women, but I'd guess the dad).
If you're born to run a business, you usually won't have any doubt about it at all, so if you are pretty sure you're capable of learning new things quickly and already have a decent foundation of math (Really anything to do with numbers), and if you're a good talker or can sell just about anything to anyone, then I can almost promise you you'll be fine without a business degree.
Also to save give a little more reassurance, I've been told by numerous people (including counselors and advisers) that business degrees are actually one of the least needed you could possibly get. As in you'd be better off taking math classes or hell, even speech classes.
In case it gives any merit, I've had 3 different businesses and generated over 30k last year on one alone. That's small time, but I'm only 20 this month. I also dropped out of HS twice, and finished out in an alternative program, so what does that tell you.
Can I ask what your family business is, if you care to disclose?