My Dad likes Inuyasha and started watching DBZ but quit when he realized that it was a "tin-plate turkey." For a long time he thought anime was junk (he'd seen Speed Racer and was unimpressed by the teenage boy with the cupie-doll eyes, and don't get him started on Sailor Moon). Then he found Inuyasha.
Now he sometimes expresses concern over my level of interest, but generally he's okay with it. Even enjoys it.
Of course, he also watches Looney Toons and enjoyed watching Blues Clues with my baby brother.
So maybe he's just a big kid.
My Mom doesn't care for animated anything much. She's a very serious lady and prefers old mystery novels like Agatha Christy's books or even older novels like Emma and Pride And Prejudice. She usually leaves the room when the family sits down to a movie like My Neighbor Totoro.
So sad.
Im getting my sis into anime, my parents seem supportive, my mom wanted to make sure Inuyasha was 12yr old friendly first even though most of her friends watch it, and she was ok with me going to anime north and is letting me go next year 
(she said dont bring me home any yaoi or Im gunna be pissed 
)
)
, but my uncle is obcessed with it.