From the Super Mario Wiki
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 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, and the character is mentioned on Princess Toadstool's profile in a 1993 character guide produced by Nintendo of America.
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 as the Japanese manual does not mention him. 1987's The Official Nintendo Player's Guide mentions that he ruled the kingdom with a queen. Although unmentioned in either the games themselves or the manual for Super Mario Bros, he was mentioned in the Japanese guidebook How to Win at Super Mario Bros..
Nintendo Comics System
In the Nintendo Comics System, the Mushroom King (also called King Toadstool or just Toadstool) is the de jure leader of the Mushroom Kingdom. However, he is depicted as scatterbrained and dimwitted, so his responsible daughter often performs his royal duties as de facto ruler in his stead. Wooster, the "chief mushroom assistant" and the king's personal subject, is occasionally at odds with him due to his lack of 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). One story also implied that he is frequently kidnapped by Wart.
Nintendo Adventure Books
The Mushroom King appears in most of the Nintendo Adventure Books, continued from his depiction in the Nintendo Comics System.