I wouldn't worry too much about her finding out, I think you are going to have to get to a place where what you want matters more than whatever silly games they are playing. This is your life, you are entitled to reach out to your father if you want, so do it. Don't try and play into their drama web, do you own thing. Your own thing is you ring up your dad, say "hey dad, been a while, how are you," and let it go from there. If you mom gives you crap, you just tell her the part about you should have the right to talk with your own father, and will do so. I don't think she'll give you crap, but if she does, ask her to consider if someone didn't want you to speak with her, would she be cool with that.

You could always text him and tell him you'd love to chat and ask him what a good day/time to call him would be. That way he's not receiving your phone call out of the blue at a time that might catch him off guard.