My recollection is fragmentary, but the slant is different.

My uncle must have been the one to quote the lyric in an adult conversation with my mother back in the '40's or early '50's. I later asked my mother what the rest of the lyric was, and she refused to tell me. That made it stick in my mind.

Phil was a tool maker in San Francisco, and married only later in his life. His primary recreation was nightclubs and bars. He developed a wide acquaintanceship with the owners, bartenders, and entertainers. After a road trip in 1952, he took me out one night to four or five places, and I got to meet many of them. We had a great dinner at Grison's on VanNess, saw some shows in small places on Broadway, and even went backstage to talk to some of his friends. . . . and there's the connection. Both of the shows I remember featured parodies of well known songs. i was especially struck by showgirls singing naughty versions of children's rhymes. . . and we visited them in their dressing-room!

That fireman's song is ripe for parodying. Has anyone got some old "party records"?