I can sort-of relate.

When I built up the courage to tell my parents about my Sexual Aversion, they honestly sounded like they would rather I had told them I was gay or bi. Especially my Dad. He makes no secret of the fact that he doesn't understand, doesn't brother trying to understand, my feelings. To him, I'm just in an odd transitional, rebellious phase because how can I hate something I've never done? He means well, and he doesn't try to force his opinions on me, but his refusal to accept me still doesn't feel great.

But in the end, it's us as individuals who define and decide who we are. While our parents approval and trust would mean the world to most of us, it's not necessary to building a sense of identity. So you just keep on being who you are.

As for the whole finding out about your girlfriend via searching your phone. That sets off a few alarms, that's seriously uncool. I'd personally be more mad about this blatant invasion of privacy than them finding out about your dating preferences.
It might be hard, I don't know what kind of relationship you have with your folks, but I'd suggest talking with them about it. Tell them it's not "just a phase," tell then the jokes hurt your feelings even if they are trying to make you feel more at ease, but also have a serious discussion with them about your own privacy. Ask them why they felt the need to go through your private phone messages instead of just asking you.

Best of luck.