My parents still can't quite grasp that I like 60s-style music. My mom especially. Even the more recent stuff I listen to tends to have an "older" vibe. Though I wouldn't say older, only that the elements I like are not currently in fashion. (I love songs that emphasize melody/harmony, and which mess with chord structure in creative ways. The second one has all but vanished from pop.)
I think it depends on a couple things: What kind of music is the kid naturally drawn to? And do the parents push "old music" very hard? 'Cause kids are always gonna rebel against something foisted on them, even stuff they'd otherwise enjoy.