My mother liked swing and big band; my dad was into Appalachian music (Carter Family, etc.) and old-time country/folk (Johnny Cash, Woody Guthrie, Mac Wiseman, etc.). Yeah, I picked up on that. Dad played guitar (at age 87 he still plays, and has his own group) and got me started as a player. He bought a Wurlitzer jukebox (78 RPM) when I was a little kid, put it in the basement, and filled it with stuff by Elvis, Bing Crosby, Lefty Frissel and folks of that era. So even before I was old enough to handle a record player, I could play music when I wanted to just by pushing the buttons. Later on, when I veered off into jazz, rock and R&B they were very tolerant, even encouraging.
My own kids (now adults with families of their own) definitely dig a lot of the same music I do. They're not as much into classical as I am (yet), but both love great rock, blues, R&B and jazz. They each branch into their own areas, but the core of their own collections are the albums we had around the house when they were growing up. We took 'em to a lot of concerts when they were young (Allman Brothers, Santana, George Thorogood, Arlo Guthrie, whatever) and also to local jazz clubs to hear small combos up close. My son took up guiter & bass (along with a brief stint on trombone in high school), and my daughter went with the sax. The cool thing was seeing them come home from the library with albums they had discovered on their own, exploring and digging deeper into the music.