I really should start a blog on battling insurance companies, I've done it enough
I've gone through the exact same situation, turning 21 and losing my parents' insurance while having UC. (Or maybe it was 22, I can't remember. . depends on your state)
You call up your dad's insurance company, and on the phone menu there should be a button to mash for COBRA service. You explain to them that you're going to lose coverage at your next birthday, and they will send you paperwork. You fill out the paperwork, and you will have to pay whatever your dad was paying for your coverage, plus a certain percentage over that. You keep all the same doctors, copays, coverage, etc.
Now, that gives you coverage for a limited time, I forget exactly how long. I think it is 3 years for "Loss of Dependent Child Status."
The thing is, it won't last forever, and it won't help you get individual coverage on your own. What it WILL do is this: When you get group coverage either by your employer, or spouse' employer, or whoever, your UC doctors' visits and medications will be covered right from the beginning. If you do not have COBRA extension, if you have any gaps in your coverage, when you do end up getting group coverage somewhere, they won't cover your UC for the first year, because it's a "pre-existing condition." If you have maintained coverage at all times, they MUST pay for it right away, by law.
Disclaimer: Your state may have more benefits, I'm just talking about
the federal COBRA law.
Here's the Department of Labor's PDF file on COBRA: http://www.dol.gov/ebsa/pdf/cobraemployee.pdf