The entire bathroom rap, especially when you enter Cool mode. It's so hilarious that you could forgive the developers for making it That One Level.
Or when you fail, in which the Potty Failure is depicted as a rocket taking off inside PaRappa, as a way of indicating that he pooped his pants.
The cutscene is even worse.
"I need to go/just as bad as you/What I had this morning I don't even want to say to you/Kick, punch, turn, and chop the door!"
In the Stage 7 cutscene in UmJammer Lammy, Lammy goes through a number of flashbacks as she rushes to the MilkCan concert. One of them involves Katy trying on different costumes and asking for Lammy and Ma-San's opinions on each one. Lammy is too busy working with electrical equipment, and Ma-San is reading a book and doesn't care about what Katy wears. Lammy then plugs in the electrical cord, electrocutes herself, and the electrical surge results in her hairstyle turning into an afro, which Katy and Ma-San really like.
Katy: Lammy, that's it!
In PaRappa's side-story in UmJammer Lammy, P.J. Berri's definition of "rock n' roll".
Another from PaRappa's side story in UJL. He, PJ, and Sunny are helping with Milkcan's performance in various ways. Sunny's helping to design costumes and shows one to PaRappa. He then requests to 'see it from the side...'
A third one from PaRappa's side-story, in Fright Flight, you get this nice little subversion of the typical call and response format of the game:
Fussenpepper: I'll miss my train. PaRappa: You'll miss your train. Fussenpepper: I miss my brain. PaRappa: May I help you?
The Rappin' Cool freestyles can certainly turn into this:
Put CRACK CRACK CRACK X X into the bowl!
If you listen very closely during the ending of the last rap in PaRappa 1, you can hear...
Joe Chin: I really appreciate this fabulous party you all had just for me!