Well, it sounds to me as if your Dad has made his limits very clear.
That doesn't excuse his homophobia, at all, but no matter the source, when a person makes clear that they simply are not open or interested in hearing about others' sexuality, IME, the best tack to take is to respect that and not force that information on them.
Perhaps it would be best for you to simply start by confiding in your mother alone.
FYI, as I said to you in the Biosphere advice letter you sent, you get to choose whatever labels for yourself you like. However, it goes without saying that any label to descibe a persons gender orientation is not likely to sum up the whole of their sexuality or sexual identity, so there's no need to feel you have to choose one word which does that -- I'd be amazed if one word summed up the sexuality of even the most vanilla, heterosexual, mainstream person there was in the world.
Sexuality is a vast spectrum for everyone, queer or no, and we should feel no obligation to try and sum it all up objectively in a word for anyone.
And also FYI, "pansexual" is the most common term those of us use to explain that we are attracted to more than binary gender (male and female), and to include other biological genders (such as intersexed people) as well as other gender identities and constructs (such as transgendered or genderqueer people). And for some, gender may just be a nonissue.
Editor and Founder, Scarleteen
My epitaph should read: "She worked herself into this ground."
-- Kay Bailey Hutchinson