From the Super Mario Wiki
Z Koopas are enemies found in the game Hotel Mario. They look like Paratroopas, but wear white shirts and black swallowtails over their shells and sport bow ties. Due to their formal style of dressing it can be assumed that they work at Wendy's Blitz Snarlton Hotel. They also appear often alongside Bellhop Goombas.
Z Koopas are mostly prominent at Wendy's Blitz Snarlton Hotel and they make their debut appearance in level 6-6. Since they are airborne they can move freely between the different floors of a level. Z Koopas appear either on the left or on the right side of the screen and proceed to travel from one side to another in a zigzag pattern. Mario or Luigi can defeat them with a well-timed Jump or a Fireball.