That is a very difficult situation to be in. The other responses in this thread seemed very helpful, and I would also recommend reading the following article:
Lastly, if I were in your place, I would want to understand why your parents refused to fill out the FAFSA documents. Once you understand their concerns, do your best to prove that they don't need to worry. If it's about privacy, tell them the information is secure and completely confidential. If they're afraid that they will have to pay your loans for you, ask your finance counselor to only give loans your parents do not have to co-sign for.