We told you in an earlier thread
Where scholarships fall short, get financial aid (if you haven't completed the FAFSA yet, what are you waiting for), and where that falls short, borrow. You do not have to pay back student loans until you finish college.
There is lots of free money for college, and FAFSA (or your local state equivalent) is your first step. It is essentially an application for financial aid (grants and GSLs), which you and your parents must complete before you can get any federal (or state) aid.
Guaranteed Student Loans are loans you take out and you have to pay back. Repayment is deferred until you are out of school. But you are responsible for it.
It sounds like the loan your parents are thinking of is not one of these loans. In fact $20K with repayment at $800 a month sounds like a five year loan with a pretty steep interest rate. If your parents are willing to take responsibility, then you should accept it, but carefully consider the repayment terms they want. It may be unlikely you could live up to their expectations ($800 a month is A LOT) in the time frame they are expecting.
The nice thing about GSLs is that even after you graduate, if you can't make the payments, you can take a few courses at a local community college and extend deferment on the repayment. And the other nice thing is that you have no obligation (financial, or other strings) to your parents.
You really must go talk to your guidance counselor at your school about this. Park your rear outside her office door if necessary. The longer you wait, the less money there is (filing date for FAFSA opens on March 1 each year, I think).
Hope you like Top Ramen (that's how I made my student loans last, and got them paid off quick after college).
Good Luck, and get cracking.
Shoshana