Junior's mom and dad
From the Super Mario Wiki, the Mario encyclopedia
Due to their limited appearance in the episode, little is known about them. The mother appears to be the dominant parent of the two, as she does almost all of the talking (the father only has one line). Although they care about their son, they're secretly desperate to get some time away from him.
Later on, when the mom and dad come back, the mom finds Junior taking a bath. She assumes that Mario and Luigi did such a good job babysitting her son, that she desperately asks them if they're willing to babysit the following Friday.