From the Super Mario Wiki
Mrs. Periwinkle is an elderly friend of Mario and Luigi's, and was presumably a neighbor of theirs while they operated Mario Brothers Plumbing. Mrs. Periwinkle never had a son of her own, and wound up treating her goldfish Kenneth as one instead. One of Mrs. Periwinkle's hobbies is horseback riding.
In The Super Mario Bros. Super Show! live-action segment "Goodbye Mr. Fish", Mrs. Periwinkle arrives at Mario Brothers Plumbing, where she is greeted by Luigi, whom she asks if he can look after Kenneth while she goes horseback riding. When Luigi denies Mrs. Periwinkle's request, she turns to Mario, who agrees to watch Kenneth just because he cannot say no to her. Before Mrs. Periwinkle left, she reminds Mario that Kenneth is like a son to her, and warns him not to listen to Kenneth if he wants for a piggyback ride.
While Mrs. Periwinkle is gone, Mario accidentally kills Kenneth by contaminating his water with a meatball. Mario, Luigi and the Ratigator all work together to try and replace Kenneth with another goldfish, but they cannot do it by the time Mrs. Periwinkle returned. Mario is so frightened by Mrs. Periwinkle's return, that he hides under the table while Luigi tries to convice her that Kenneth swam away.
When Mrs. Periwinkle fails to buy Luigi's story, Luigi tells him to come up from under the table and confess to Mrs. Periwinkle. Mrs. Periwinkle demands to know what happened to Kenneth, but before Mario can speak, the Ratigator gets his attention, and directs him to a replacement goldfish that he found. Mario presents the new goldfish to Mrs. Periwinkle, which she assumes is Kenneth. Mrs. Periwinkle is so pleased to have "Kenneth" back, that she offers to pay Mario for looking after him, but he refuses her offer. Before leaving Mrs. Periwinkle tells Mario and Luigi that she will recommend them to all of her friends who need fishsitters.
She is briefly mentioned by Mario in "Defective Gadgetry" when he discovers the original Kenneth in a bucket of water.