Marcia and Jon Pankake's book, Joe's Got a Head Like a Ping-Pong Ball - A Prairie Home Companion Folk Song Book, has an index where the songs are arranged by tune. The most popular by far is BHotR. The songs/parodies listed under this heading are

Birmingham's My Home
The Bugs Marched Down the Aisle
The Bulldog and the Bullfrog
Chicken Sandwich
Emily Dickinson
Glory, Glory, How Peculiar
Glory, Glory, Pork Superior
Mine Eyes Have Seen the Glory of the Burning of the School
Mine Eyes Have Seen the Horror of the Ending of the Term
My Pink Pajamas
Oh, Ay Liff in Minneapolis
She Waded in the Water