My parents detested anime when my brothers and I first got into it. Why?
...well, I mostly blame it on the fact that our first anime was Pokemon, and that we were obsessed with it to a barely sane level. >.< I think in general they just got tired of hearing about Ash's latest adventures over the dinner table. (In our defense, we were respectively twelve, ten, and six at the time.)
In the years since then, I have gradually won them over to tolerance, acceptance, and even grudging approval. How? Well, first, I tried to shut up about it, since it was constantly hearing about it that messed things up in the first place...and second, I tried to gently introduce them to parts of the fandom I thought they would like.
My father, an openminded person, drove me to my first conventions and liked the camaraderie he saw there; he was also proud to stand outside the Glacier Room at Sakuracon '04 and listen to the crowd cheering his little girl's first public filk. (He didn't have a con badge, so he couldn't come inside.) I finally got his approval of a few carefully chosen series, like Dragon Half and Cowboy Bebop, by asking him to watch them with me and making sure they were the kind of thing he likes.
My mother was a harder nut to crack, but she finally agreed that it was a good hobby when I started cosplaying. She adores sewing and was delighted to have a reason to teach me. Thanks to that, I'm a competent cosplayer and have won prizes for my work, and she's finally approved my fandom, if only because it led to my becoming a seamstress.