Mario's parents are shown in Yoshi's Island, among other related anime and comics.
At the end of Super Mario World 2: Yoshi's Island, Mario and Luigi's mother made her first video game appearance. She is portrayed as a wide-set woman wearing a pink nightgown, and only her lower half and hands are seen.
The video game's artists portrayed him as a skinny man in a nightshirt of indeterminate color. Only his lower half and hands are seen.
The Mario Wiki has articles on both Mama Mario and Papa, although there are a lot of references to the live action Super Mario Super Show, which is probably not canonical.
Likewise, the story is told differently in the 1993 Super Mario Bros movie, where the two are orphaned when Luigi is still a baby. Again, there were major rewrites to canon for the movie.