The footage of Lisa calling Homer at work to say she loves him is recycled from Bart's Dog Gets an F (1991), as you can clearly see Lisa has the mumps in this scene.
One of the children at the talent show performs the song "My Ding-a-Ling" by Chuck Berry. According to Al Jean, it was a "huge difficulty" to clear the rights for the song so that it could be used on the show. John Boylan, who produced the album The Simpsons Sing the Blues, personally appealed to Berry to clear the song for them. The lyrics to "My Ding-a-Ling", with their sly tone and innuendo, caused many radio stations to ban the song. This is parodied in the episode when Principal Skinner rushes the child off the stage before he is able to finish the first line of the refrain as ding-a-ling is slang for male genatalia.
Al Jean and Mike Reiss came up with idea for the episode while going through a list of Lisa's interests, and Jean told Reiss, "Lisa likes ponies; we [should] give her a pony." While writing down ideas for the story, they decided to explore the consequences of having a pony in a suburban house.
In this episode, Ralph speaks with Nelson's voice