Nate dresses as Ellery Queen, the detective played by Timothy Hutton's father, Jim Hutton in a 1970s TV series: Ellery Queen (1975). He even wears the hat his father wore in the role, and refers to the character as the greatest American detective.
The band at the murder mystery party plays theme songs from various mystery movies and TV shows throughout that part of the episode.
Sophie uses the fake company name "Mystery, Inc," which was the name of the team from Scooby Doo, Where Are You! (1969).
The client says that she was an employee at the law firm of "McGann, McCoy, and Baker". Colin Baker, Sylvester McCoy, and Paul McGann were the last three actors to play The Doctor in the original series of Doctor Who (1963) and the TV film Doctor Who: The Movie (1996). And there was also the fourth Doctor, played by Tom Baker.
Parker's first grab may be a reference to Imelda Marcos. Imelda Marcos, who is the widow of a former Philippine president, is well-known for having a large collection of shoes.