The joke where Peter is smothered in the Alpha Delta Pi sorority pile of girls clapping and chanting is based on the real sorority's recruitment video and chant that went viral on social media in 2016.
When demonstrating she actually has the gymnastics skills that Peter and Lois fabricated to get her into Brown University, Meg injures herself and wants "Dr. Nassar", until a lady whispers something to her and Meg says, "Oh my stars!". Dr. Nassar is former USA Gymnastics team doctor Larry Nassar, was convicted of sexual assault on female gymnasts he was treating and has at least 265 victims from about 1992-2015.
In the scene in the living room, from outside the house, we hear Cleveland sing a line "Mail time!" in different keys at the same time and Peter yells out, "The mail's here!" Then Peter breaks into song. The song sung by Peter is an actual song called "Mail Time" from the Nickelodeon interactive kids show Blue's Clues (1996). This is a reference that whenever the host hears "Mail time" he ends up singing the song before getting the mail from the mailbox character who enters the room through the window.
When Brian tells Meg she can stay at Brown if she can prove she can do all the things Lois and Peter claimed she could do on her falsified application, Peter says excitedly that he can "get back to hustling drunk women upstairs with Tobin and Squee". This is a reference to the Brett Kavanaugh scandal where allegations of sexual assault lodged against Kavanagh--who had been nominated for a seat on the US Supreme court--were dismissed by his advocates as beer-induced frat-boy behavior with his friends Tobin and Squee.
The plot of the episode is based on the 2019 college admissions bribery scandal.