New year's police

The New Year's police are a law-enforcement organization that, according to Francine, monitor that at midnight on New Year's Day, everyone throws out their old calendars. If anyone doesn't, they know, and they come to arrest that individual.

In a fantasy sequence of Arthur Read's, they come to arrest Grandma Thora. They are depicted as wearing party hats.

After hearing hearing of this from Arthur, Binky asks if he's "goofy" and tells him that there are no New Year's Police, but rather a New Year's wrestling match.

As Grandma Thora was not arrested by these police after failing to throw out the old calendars immediately, it is extremely likely that the New Year's Police were just a tale that Francine had invented or heard.