From the Super Mario Wiki
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The Mushroom King is a seldom-seen ruler of the Mushroom Kingdom, and the father of Princess Toadstool. Although he is said to rule alongside the princess, he is a figure that is mostly absent in the video game series. However, he does make somewhat notable appearances in other early media such as the Nintendo Comics System and Nintendo Adventure Books.
Super Mario Bros.
In the NES instruction booklet of Super Mario Bros., the Mushroom King is briefly mentioned in passing. With few exceptions such as Super Mario Bros. Deluxe, this is generally not included in subsequent releases. This is likely because the Japanese manual does not mention him.
Nintendo Comics System
In the Nintendo Comics System, the Mushroom King is the de facto leader of the Mushroom Kingdom. However, he is depicted as scatterbrained and dimwitted, so his responsible daughter often performs his royal duties in his stead. Wooster, one of his subjects, is occasionally at odds with him due to his intelligence.
Based on his role, it is likely that he was based off the kings from Super Mario Bros. 3 as he plays a similar role in the comic strips (having a magic wand that is often taken by Bowser's forces and being transformed into various animals).