The two clownfish at the beginning may be a reference to Marlin and Nemo from Disney/Pixar's Finding Nemo and its sequel Finding Dory.
Although it is unclear how long the film takes place after Despicable Me 2, it can be reasoned that very little time has passed between the two movies. There are several pieces of evidence that support this claim such as the fact that the girls surprised Gru and Lucy with a Hawaiian luau style party for their "honeymoon". Additionally, Lucy's personal conflict within the film was discovering herself as a new mother to the girls, suggesting that a very small amount of time passed between the end of the second film and the beginning of the third film.
This film marks Trey Parker's first role in a family film.