The Anniversary Waltz is the name it's known by in Jewish circles. It was made famous by Al Jolson who was, to audiences of the 20's and 30's, what the Beatles were to the 70's. My father sang it when he drove our first car, a 1948 Nash with front seats that dropped back to make a bed. I often fell asleep while he sang that song. We didn't start singing the parody until we were in High School and more "sophisticated".
He also sang "Big City Blues" and a novelty tune, "Did You Ever Hear Pete Go Tweet, Tweet, Tweet On His Piccalo?" ("...Well, I'll put you wise/He's a bird in disguise/A bird called Piccalo Pete)
Everyone in my family knew the Anniversary Song. It was played at all weddings as the first dance.