My dad's totally stuck in the 80s, with his old REM and Oingo Boingo, it's cute. The two of us have music-swapping nights and listen to the Pogues and the Cure, and I let him burn all of my CDs. He really likes The Decemberists and The Arcade Fire. I think it's the most adorable thing ever.
When my mom and I go on road trips, we listen to Billy Joel and soundtracks from different musicals, or John McCutcheon, James Taylor, John Denver. She can sing along to my favorites from when I lived back at home, like Weezer and Rilo Kiley and Ben Kweller, but it sounds like she's singing folk songs to little kids no matter what. I can't tease her though, because I make everything sound like opera.
My parents are pretty tolerant of my taste in music, although neither of them let me play Joanna Newsom when I'm around them. My dad says she sounds like a midget riding a pony.