I do believe that both parents have to be with the teen. If I were the non-custodial parent, I'd want to know for sure why my child has a passport. I'd also want a say if I found out that my ex was considering a trip to Uganda. Uganda is a scary place and it wouldn't sit well with me to not have some say in the matter, regardless of whether I had primary custody or not.
I would NOT try to do an end-run around your ex. (Not saying that you would, mind you. However, just by the fact that you are asking if the father has to sign, I'm a little concerned.) I'd work on getting his agreement so that he doesn't find out and take you to court to prevent it.
Dawn