They're not always taught by their parents, according to this Seattle study of song sparrows. Adults don't sing extra slowly in front of their chicks, but young birds do seem to listen to adults, and somehow end up learning a song from certain grown-ups while ignoring others.

Young birds memorize what a grown-up song sounds like.  As he learns, his song is nonsensical, like human baby-talk.  Gradually the young bird can produce recognizable versions of what he heard from the old-timers.  But birds pick up new songs over time throughout their lives.