Hey, if we’re going there, then: Paul McCartney, singing about “Silly Love Songs”.
“Merry Christmas from the Family” includes fragments of “Feliz Navidad” and “Silent Night” with a reference to “The First Noel” played on the stereo.
The whole point of “Rock And Roll Heaven”, by the Righteous Brothers, is to talk up other singers and their songs: Jim Croce and Bad Bad Leroy Brown; Bobby Darin and Mack The Knife; and so on for Janis Joplin and Otis Redding and et cetera.
Other than Killing Me Softly, I’m not sure most of the subsequent examples really fit what the OP is asking about. The question seems to be looking for examples of songs that are about another song, not just that mention or list other songs.
Hootie & the Blowfish’s “Only Wanna Be With You” references Dylan’s “Idiot Wind” and “Tangled up in Blue”:
Elton John’s Goodbye Yellow Brick Road.
Garden Party, by Ricky Nelson:
Hello Marylou was a Nelson song. She Belongs to Me is a Dylan song that Nelson covered.
I think this one qualifies:
So does Sing by the Carpenters.
My mention of “Going for a Song” certainly qualifies on both counts; it’s a song about having to play “A Whiter Shade of Pale” night after night after night so that you’re sick of it.
Genesis “the musical box”
George Jones “Who’s gonna fill their shoes?” references a ton of other songs and artists George Jones - Who's Gonna Fill Their Shoes? - YouTube
Lyrics at Google Play Music is no longer available
Seeing this thread reminds me very much of one from 10 years back. Not identical, just a lot of overlap. Yeah, I got a lot of free time.
Sort of a tangent to the OP, but regarding referencing other artists I’ll add John Hiatt’s “Memphis in the Meantime”:
Come to think of it, Hiatt references his own song in the song.
Paxton also sang “Has Annie Been In Tonight?” and “When Annie Took Me Home.” Being a huge fan of his music, I love the tunes.
Perez Hilton’s Thank You Broadway incorporates the titles of songs from various shows in the lyrics. Really well done.
How about Sweet Soul Music, by Arthur Conley? Several references, such as:
When Annie Took Me Home has the lyric:
Rafters rung, songs were sung
Like Spanish Is The Loving Tongue
Thank you Doug, you are correct.
I also meant to specify: NO (MORE) AMERICAN PIE!
That one qualifies. What book is that – it sounds interesting.
“Vera” from *The Wall *is about Vera Lynn and her signature WWII hit, “We’ll Meet Again.”
Saint Dominic’s Flashback: Van Morrison’s Classic Album, Forty Years On
Yeah, I know it’s clumsily titled, and a bit short, but it gave an interesting account of the various sessions when the album was recorded and the involved musicians and producers. It’s also kind of a snapshot of Morrison’s professional life in early 1972. It’s not a masterpiece, but very interesting, and you can’t do wrong with $ 1.42 for the kindle version.