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After Brian gets a $400 parking ticket, he realizes that Mayor West is a money scamming corrupt Mayor, and sets out to expose the truth. However, Meg, Mayor Wests new more than personal assistant, is dead set against it. Meanwhile, Peter and Lois decide to write songs to perform at Chris' talent show. They break out the marijuana for some "inspiration".
A very creative title from the Producers, because it references both plots of the episode. First, is the obvious. "Deep Throat" was the codename of a man who recently backstabbed the Nixon Administration by giving out top secret information on the Watergate Scandal. This of course is referencing the scandal Brian is trying uncover concerning Mayor West. The second would be that "Deep Throat" is also the name of a form of singing which involves the vibrations of the diaphragm. Which references Peter and Lois' singing in the episode.
The little red creature that hops around Mayor Wests office is called a "Noid". The Noid was the mascot for Domino's Pizza in the 80's, and was a person in a red bunny suit with an "N" on his chest. He used inventions to destroy people's pizzas. Mayor West states "Perhaps the Noid should have avoided me", which is a spoof on the Noid's catchphrase "Avoid the Noid".
Burger King Christmas Carol
When Peter flashes back to when he worked at Burger King; and we see him singing a carol is a spoof of a video that passed around the internet starting in 2002. Family Guy has been known to spoof these internet videos,as when Brian dressed up as a banana and sang "Peanut Butter Jelly Time" in an earlier episode.
Black or White
The part where Cleveland is on top of Brian's car smashing it; and then jumps off and turns into a Jaguar is referencing Michael Jackson's video "Black or White", in which he did the same thing.
Teenage Mutant Ninja Turtles
You can hear the theme song to the classic show "Teenage Mutant Ninja Turtles" when Mayor West and Meg are watching TV.