The cutaway I am invoking is from the Family Guy-Simpsons crossover from a few years back. The problem is, it was created after South Park did their Family Guy episode--out of spite, of course, and completely unprovoked, too.
(Scene: The interior of the family Guy main studio, facing the glass wall behind of which several manatees are nosing floating balls from a big netted collection of balls toward a chute.)
(Seth MacFarlane is watching the manatees at work. At the bottom of the chute is a view-box receptacle wherein five balls have just descended. Seth MacFarlane reads off the five balls.)
Seth MacFarlane: Ipecac Contest, WWI Air Force, Homer Simpson, Bob Belcher, Cleveland Brown." Sounds like a Family Guy joke!
(Cutaway back to the living room where Peter, Brian, Chris and Stewie are having their Ipecac drinking contest which induces a considerable amount of "purging.")
(The puking finally enters a lull and the four appear to be recovering. Lois enters the living room with a big bucket of clam chowder.)
Lois: Okay, who wants chowdah?
Peter, Brian, Chris and Stewie (all puking in unison): Wruaaaugh!
Lois (puts the pot back in the kitchen and turns to the four men): Peter! You and the boys have thrown up all ovah the flooah! And look! You even have vomit on the furnichah -- and all over yourselves, too!
Peter: You think that's bad! Remembah the time I flew the in the World War One Air Force with Homer Simpson, Bob Belcher and Cleveland?
(Cutaway to the scene in the Family Guy - Simpsons crossover show where Cleveland goes down. Again.)
Homer: Woo-hoo!
Peter: Yeah-ha-haah!
Bob Belcher: (popping up from the rear of the plane): Yeah, we did it!
Homer (angry): What's he doin' here?
Peter (non-chalant): Oh we gotta carry him; he can't fly on his own. We let that other guy try, and look what happened.
(Cutaway to Cleveland's plane, the engine of which just caught fire.)
Cleveland: No, no, no, no,NO!! (as his plane enters its death spiral).
Okay, this is my last South Park Family Guy joke. I've run out of ideas! 😁
SOUTH PARK FAMILY GUY IN THE ROAD TO THE MULTIVERSE
Seth MacFarlane really should have cut a deal with Trey Parker and Matt Stone, the creators of South Park, to include their version of Family Guy in the first installment of "The Road to the Multiverse."
Here's where I envision it to have been placed:
(Conclusion of the Disney universe clip as Mort Goldman is being beaten up by the Griffin family)
Stewie: Oh, but look how shiny my buttons are here!
Brian: Just push the button!
Stewie: Yeah, yeah. Okay, I'll push the thing. (Stweie hits the button.)
(Bruce the fairy appears, waves his magic wand, causes Stewie and Brian to disappear, then disappears himself in an explosion of stars just as Mort Goldman falls to the floor, unconscious. Brian and Stewie then reappear in the same spot in the South Park version of Family Guy.)
Brian: What the-?
Stewie: According to my readout, everything here is drawn by the team of Trey Parker and Matt Stone. This is the South Park universe, Brian.
Brian: Which explains why everything is drawn on the cheap, like on a Commodore 64.
Stewie: And the strange sound of our voices.
Brian: Except for yours. (Stewie gives Brian the dirtiest look, then looks back down at the machine.)
Stewie: It also means all the jokes are written by manatees!
(Cutaway to the interior of the main office of the Family Guy Studios, facing a glass-walled pool with a huge, netted collection of small beach balls, several manatees and a chute into the receptacle within the studio itself. Seth MacFarlane watches as the manatees nose five balls into the chute. The balls descend into the receptacle, each bearing one word.)
Seth MacFarlane (reading the word on each ball in succession): "Laundry," "Sing" "Variety Show" "Paul McCartney," and "The Cockroach Song." Sounds like a Family Guy Joke!
(Return-cutaway to the Griffin family sitting in their living room. The South Park universe's Stewie and Brian are with the rest.)
Lois (with a rhotic accent): Peter! You forgot to take the laundry out this afternoon.
Peter (in a bad imitation of Archie Bunker doing a bad imitation of Peter): You think *that's* bad--remembah the time I sang LaCucaracha for Pawl McCaatney?
(Cutaway to multiverse-travelling Stewie and Brian standing behind the couch.)
Stewie: I'm not standing for this, Brian.
Brian: Me neither!
(Stewie hits the button. Brian and Stewie disappear and then reappear in the Robot Chicken universe.)
Brian: Woah, this is trippy!
Stewie: I should say so; we're in the Robot Chicken universe.
Peter (in anger): Will you guys move? You're blocking the teavee.
And so it goes, with only about 70 seconds to be cut out of the Dominant Dogs universe at the end of the episode to make room.
But it was not meant to be because the creators of South Park despise Family Guy! 😠