August 07, 1992
Near the beginning of the movie, the three kids are seen in their room, with Rocky talking to his friend Emily with a tin can telephone, Colt is bouncing on a trampoline and Tum Tum is playing Supe Mario Bros. 3 on the NES. Specifically he's seen playing in World 5-1. After a while, as the three hurry in their beds not to be cought still awake by their mom, Tum Tum is seen turning off the TV, with Super Mario Bros. 3 title screen on.
May 06, 1994
In a scene in the movie, a giant balloon shaped like Mario can be seen.
It's unknown if Touchstone Pictures got the authorization from Nintendo to use a Mario game in their movie, but it's possible that they didn't need one to show so little of the game.