Bob and Ray used to do a Godfrey spoof character; I think they called him Arthur Sturdley, and played him as the egomaniacal oaf he apparently was off-the-air. Stan Freberg made a record parodying Godfrey, called "That's Right, Arthur," but Capitol refused to release it for fear of Godfrey's lawyers. (I don't know offhand if it's since been released in any of the Freberg retrospective CD sets.)
The B side of Eddie Lawrence's "The Old Philosopher" had "King Arthur's Mines," where the gag at the end had King Arthur turn out to be Arthur Godfrey.