From the Super Mario Wiki
“This looks like a Penguin Murder Mystery!”
The Penguin Patrol is a green Bumpty and an inhabitant of Shiver City, where he works as a police and seemingly the city's only law enforcer. Unlike all the other Bumpties in the city the Penguin Patrol is green.
In Paper Mario, the Penguin Patrol can first be encountered near a frozen pond in Shiver City. If Mario tries to Ground Pound (or explode with Bombette) the ice that has formed over the pond, the Penguin Patrol will stop him and reprimand him for risking the lives of nearby skaters on the pond, who would have presumably drowned in the pond if Mario successfully smashed through it.
Later, after Mario is found near the seemingly dead mayor of Shiver City, the Penguin Patrol is brought to the mayor's house by the mayor's wife to arrest Mario for murdering Mayor Penguin. Instead of arresting Mario though, the Penguin Patrol, after hearing Mario proclaim that he was innocent, decided to allow Mario to prove his innocence, though the Penguin Patrol alerted the populace of Shiver City to be wary of Mario and not let him leave the city. While the Penguin Patrol is at the mayor's house Mario can now freely break the ice over the pond to get the key that is under the frozen pond.
The Penguin Patrol eventually exonerated Mario of any wrongdoing when it was discovered that Mayor Penguin wasn't actually dead and that he had, in actuality, simply knocked himself unconscious while wrapping a gift for his friend Herringway, who Mario had brought to the Penguin Patrol upon finding a piece of paper in the thought-dead Mayor Penguin's fin. After this incident, the Penguin Patrol can be seen patrolling Shiver City; Mario can speak with him, though he really has nothing interesting to say.