From the Super Mario Wiki
Ads keep the MarioWiki independent and free :)
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 arrived at Mario Brothers Plumbing, where she was greeted by Luigi, whom she asked if he could look after Kenneth while she went horseback riding. When Luigi denied Mrs. Periwinkle's request, she turned to Mario, who agreed to watch Kenneth just because he couldn't say no to her. Before Mrs. Periwinkle left, she reminded Mario that Kenneth was like a son to her, and warned him not to listen to Kenneth if he wanted for a piggyback ride.
While Mrs. Periwinkle was gone, Mario accidentally killed Kenneth by contaminating his water with a meatball. Mario, Luigi and the Ratigator all worked together to try and replace Kenneth with another goldfish, but they couldn't do it by the time Mrs. Periwinkle returned. Mario was so frightened by Mrs. Periwinkle's return, that he hid under the table while Luigi tried to convice her that Kenneth swam away.
When Mrs. Periwinkle failed to buy Luigi's story, Luigi told him to come up from under the table and confess to Mrs. Periwinkle. Mrs. Periwinkle demanded to know what happened to Kenneth, but before Mario could speak, the Ratigator got his attention, and directed him to a replacement goldfish that he found. Mario presented the new goldfish to Mrs. Periwinkle, which she assumed was Kenneth. Mrs. Periwinkle was so pleased to have "Kenneth" back, that she offered to pay Mario for looking after him, but he refused her offer. Before leaving Mrs. Periwinkle told Mario and Luigi that she would recommend them to all of her friends who need fishsitters.
She is briefly mentioned by Mario in "Defective Gadgetry" when he discovered the original Kenneth in a bucket of water.