From the Super Mario Wiki
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.
When Junior's original babysitter fails to show up, Junior's mother finds Mario and Luigi in her kitchen, and practically dumps Junior in their arms just so that she and her husband can leave the house.
Later on, when the parents come back, the mother 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.