And it’s not a theme song per se, but the opening lines to the intro of the short-lived TV show Then Came Bronson, which Frank Conniff is obsessed with for some reason, was quoted more than once.
“Where are you going?”
“Man, I wish I was you.”
In The Girl in Lovers Lane, as Carrie is leaving her house we hear the themes from The Donna Reed Show, Leave It to Beaver, My Three Sons, and just for good measure Barnaby Jones.
Found a Beverly Hillbillies theme riff. When Big Stupid is looking at his reflection in a pond we hear “Black gold, Texas tea.”
Servo sings a bit of the Family Affair theme song in the “death” host segment in Gunslinger.
There’s at least one episode where you see an explosion or fireworks and they sing part of the Love, American Style theme. This is several years before they did their actual parody of the show.
In Batwoman I they sing Shirley Chisholm’s name like it’s the theme from the TV show The Chisholms, if I’m remembering right.
I’m positive they’ve done the theme to The Dick Van Dyke Show at least once. And, of course, they’ve even riffed on their OWN theme song once or twice, at least.
Off the top of my head, I think they used it in Santa Claus Conquers the Martians when Kimar enters.