@alexmomo00,
I think you've got the right idea, to try to hit 'pause' on this one.
I know it's hard but if you care (and you sound like you truly do), then getting her into trouble with her father is a very uncaring thing to do.
Her father sounds extremely strict and old-fashioned, but she is still a minor and she still lives under his roof. Until she becomes independent of him in some manner (moving out, getting emancipation, etc), then she's kinda stuck. Fortunately, she will turn 18 in less than a year.
If she wants to have any sort of a relationship with her father (and he might have control over her access to her mother, so keep that in mind), then
your relationship with her father has got to be at least polite. Getting too pushy about things won't help in that area.
If she doesn't want to have any sort of future relationship with him, then again, she should consider than nixing a relationship with her father might bring her mother along with the ride.
It might pay for her to have you over, completely above board, to meet her family. Not as a boyfriend. As a friend. If this happens, dress well and be on your absolute best manners (as in, bring a hostess gift if you go to their house and everything. I am being utterly serious. Even if you think that's over the top, it won't be in this instance). Listen, don't interrupt, etc. You may not win her father over immediately or at all, but at least don't give them good reasons for not trusting you or her.
"Mom, Dad, this is my friend Alex."
You: "Hello, Mr. and Mrs. ___ thank you for inviting me over."