My dad listens to many different styles of music, some of his favorites would be Elvis Presley, Neil Young, The Beatles and AC/DC, but his favorite genre would be Progressive Rock/Metal, AOR/Melodic/Symphonic (in that area).
Bands like Symphony X, Threshold, Arena, Shadow Gallery, IQ, Pallas, Haken, Redemption, Royal Hunt (funnily enough, not DT), but bands like those.
Personally I share quite a lot of music with my dad, but I don't really care for that 'kind' of Prog where all those latter bands fit in, so I don't really share that side of music with him, but I share a lot of other things.
Interesting enough, my dad enjoys almost all kind of music I play in the car, everything from Agalloch and Opeth to Radiohead and Tori Amos, he has quite a wide music taste as I mentioned.
My mom on the other hand is fairly narrow, she listens pretty much only to hard rock/'traditional' metal, artists/bands like Deep Purple, AC/DC, Whitesnake, Ozzy, Metallica, Maiden and so on.
I used to share her music-taste when I was younger and listened to those kind of bands, but seeing as I don't anymore, we really don't have much in common on that front.
My mom is one of the people I know with the smallest amount of taste in music, if it isn't like those bands that I mentioned, she drops several complaints like the music being "too soft", "too heavy", "no melodies, no chorus" and so on.
She's pretty lame, she picks on pretty much all music that isn't heavy metal or hard rock.