From the Super Mario Wiki
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The Mushroom King, is the king of the Mushroom Kingdom and Princess Toadstool's father. He was mentioned in the Super Mario Bros. instruction manual, but appeared mostly in the Mario comics. In the comics, he is depicted as a scatter-brained numbskull. King Toadstool most likely ruled the Mushroom Kingdom while Peach was a baby. However, it seems that the Princess herself became leader of the kingdom after she grew up.
Based on his namesake and role, it is likely that he was based off the Mushroom Kings from Super Mario Bros. 3 as he plays similar roles as them in the comic strips (having a magic scepter that is often taken by Bowser's forces and being transformed into various animals).
 Super Mario Bros.: Peach-hime Kyushutsu Dai Sakusen!
The Mushroom King, called the "Mushroom Hermit", also appears in Super Mario Bros.: Peach-hime Kyushutsu Dai Sakusen!. He appears once Mario and Luigi discover the Mushroom Kingdom, explaining that his kingdom is under attack by the evil King Koopa, who has kidnapped and is planning to marry his daughter, Princess Peach. He also tells Mario that the only way to save the Mushroom Kingdom and Princess Peach is to find the three legendary items: a Super Mushroom, a Fire Flower, and a Starmen. He then sends Mario off to save the kingdom and has him take Kibidango along with him. He also tells Luigi that if he helps Mario, he can keep any Coins he finds during their journey. Later, he appears along with Peach after she has been rescued and reunited with Prince Haru.
 Super Mario Bros. Film
The Mushroom King, under the name King Bowser (not to be confused with the actual President Koopa), appears as a main character in the Super Mario Bros. film as Daisy's father. He had been de-evolved by President Koopa into a giant sentient fungus that had spread across the parallel world to obstruct Koopa's construction projects.
He continually aids Mario and Luigi throughout the film in their quest to defeat Koopa. Once they successfully do so, King Bowser reverts back to his human form and exclaims, "God, I love those plumbers!"