Old Time Radio Spoofs, Gags, Pranks and April Fools!

Spoofs are probably as old as language itself, dating from the times when the tribal elder told stories around the campfire. Then, just as now, the best spoofing probably took place when the elder's back was turned!
As radio drama established itself into dominant genre forms, the forms became the target of parody and ridicule. Unlike some more modern spoofs, most radio parodies are good natured and have a distinct lack of condescending tone.
Soap Operas were an early money maker for the radio networks, appealing to house wives who would use the stories to keep them company while they were doing their house work (this was a great advantage of Radio Soaps over the later TV Soap Operas- the dishes would pile up and diapers could go unchanged while the house wife was trapped in front of the TV, but with a Soap on the radio, she could let her tears fall directly into the dishwater and not miss any of the story!)
The melodramatic elements of Soap Operas made them an easy target for parody, especially the cliff hangers in the plots, the swelling organ music, and the often ridiculous but accepted in the soaps plot devices. A popular parody device would be to expand these plot devices to ridiculous extremes, like the heroine's brother's wife's cousin recovering from amnesia. Duffy's Tavern had a lot of fun with these conventions, while Bob and Rayinvented a few of their own spoof Soap Operas.
Hard Boiled Detectives and Radio Noir was another easy target for spoofing. The conventions of the detective story, which were developed in the pulp detective magazines, like the hard drinking and violent tough guy private eye, were turned around for laughs by "little guy" comics like Fibber McGee and Charlie McCarthy. Gracie from Burns and Allen has been accused of listening to too many detective shows, and the result is nutty-ness that only Gracie Allen could pull off. And Amos 'n' Andy in the big city often need to think like a detective.
Westerns were incredibly huge in movies, TV, and Radio, and were also spoofed on a regular basis. Part of this is because Westerns were a staple of low quality B-movies. The Westerns were popular with the studios because they could be made fast and cheap, but this very cheapness lowered the expectations of quality for the entire genre. But it made for great fun with examples like Jack Benny's "Buck Benny" character and Red Skelton's often sad cowboys. Fred Allen, in many ways the prototypical New England city boy, had a lot of fun burlesquing the Old West whenever he got the chance.
The one area of Old Time Radio that seems to get little parody is the Musical Comedy Variety shows. But then it can also be argued that shows like Fibber McGee and Molly, The Fred Allen Show, and the Jack Benny Program had enough laughs at their own expense that they were parodies of themselves.