My ex left me in 2009, it's now 2014. I just found out that we have a child together. Can I just sign over my parental rights if I don't want anything to do with the child. She also has a 10 year restraining order against me.

No. The only situation here in which you can voluntarily relinquish parental rights is if she is married and her husband wishes to adopt the child - if that is the case, you can consent to the adoption.

You cannot otherwise relinquish the responsibility for your child. But after all this time, she may not want you to play a role.

There is this myth that a parent can just sign over their rights to a child. But its only a myth. Only a court can terminate rights and courts are very reluctant to do so.

How did you find out about the child? Is she coming after you for support? If so, then you will be held financially responsible. You are not legally obligated to have a relationship with the child. If she is not after support then you need do nothing.

Yes, you can sign over the right to make decisions in the childs life and you can sign over custody. But you are still the father and with that comes responsibility. So if you wish not to see your child you may do so but you still have to pay child support.