NOOOOOOOOOOOOOOOO! Now I have to go wash my brain out with Granny’s Lye Soap for the mere thought of the hint of the glimmer of what that might sound like!
Bloodhound Gang has many possibilities. In their song, “Your Only Friends Are Make Believe” off One Fierce Beer Coaster, they sing, to the tune of “Hungry Like the Wolf”:
Your best friend is you
I’m my best friend too
I share the same views
And hardly ever argue
Eat Spam from the can
Watch late night C-SPAN
And rock out to old school Duran Duran.
“Rocky Raccoon” was not the first Dylan parody. In 1966 Simon & Garfunkle did a very broad, very obvious spoof of Dylan with their rant song “A Simple Desultory Philippic.” Complete with harmonica. As if it weren’t already obvious enough, they made it more explicit by saying “When you say ‘Dylan’ <blowing harmonica> he thinks you’re talkin’ about Dylan Thomas—whoever he was. The man ain’t got no culcha!” In the fadeout, Garfunkle exclaims “Folk rock!” and then mutters “I dropped my harmonica, Albert” (a reference to Dylan’s manager Albert Goldman).
This reminds me of that Trivial Pursuit question about “koo-koo-ka-chew” appearing in “I Am the Walrus” and “Mrs. Robinson”.
On the Indigo Girls’ “Cedar Tree”, they strum just a little bit of Beethoven’s Ninth (Ode to Joy) in the instrumental part at the end.
I know there are others in the spirit of the OP itching the back of my mind but I can’t get there 
Obscure one, but brilliant nonetheless for the lucky few that know it 
Mary’s Danish do a great cover of “Foxy Lady” but break into a sneaky riff from Led Zep’s “Kashmir” during the solo, it comes across so natural you are hardly aware of it until after several listenings.
Ah, but the definitive Dylan parody still has to be the Race Marbles’ “Like a Dribbling Fram”, which also has an “I dropped my harmonica” line near the end, but leading to a far weirder conclusion.
The horror, the horror…
Dead Can Dance did a song titled “Indus” that used Indian musical motifs. They were accompanied by a lugubrious string orchestra that played an Indian melody as slow and thick as molasses. The first four notes of that melody were copied verbatim from the Beatles’ “Within You, Without You” from Sgt. Pepper, which was also played by a string orchestra. Dead Can Dance duplicated that ascending motif from George Harrison’s song as exactly as possible, to pay a tribute to the man who singlehandedly introduced Indian motifs into Western pop music.
I’m surprised no one has mentioned John Williams subtly working the Imperial March into Anakin’s theme for Phantom Menace yet.
How could I forget that one! I listened to the L.A. album only last month:
(singing)
Beef pies! He was born near the beef pies,
Underneeeeeth, Joni Mitchell’s autographed picture
Right besiiiiiiide, Elliot Robert’s Big Band Book
Next to the boooooat where Crosby flushed away all his stash
And the cooooooops got hime to the boat and drove away
To the cannnnnn, where Neil Young sliped another disc.
Heh, heh, heh, heh. God I miss Frank.
In the musical Rent, one of the characters keeps playing a particular bit on his guitar in an attempt to write a song. The bit is actually “Musetta’s Waltz” (or something like that) from La Boheme–which Rent is loosely based on. They even mention this at one point–"Oh, that doesn’t remind us of “Musetta’s Waltz”.
Dr. J
“More” sung by Madonna in Dick Tracy - the whole song is an answer to the question asked in “I Got Rhythm”