The first big, annual, bad-writing contest I could find began in 1977 and ran to 2005. Originally a promotional even for Harry's American Bar & Grill, The International Imitation Hemingway Contest, also known simply as the Bad Hemingway Contest, was a runaway success, prompting submissions from amateurs and famous writers alike. Here's an example from the 1986 winner, The Snooze of Kilimanjaro by Mark Silber:
Now he would never write the things he had saved to write until he learned to spell them. For instance, accommodation. One C, two Ms, or the other way around? He wasn’t sure. Or chrysanthemum? On rugged Kilimanjaro, there was not even a dictionary.
“How do they know I’m dying?” he asked the woman, indicating the crowd of undertakers, florists and wake caterers who were gathering at the edge of the campsite. “Is it the odor?”
“Of your poop?” she asked. “Or your poetry?”
She knew how to hurt him this woman, this female being, this person of the nonmale persuasion. And he would have hurt her back, at least challenged her to a thumb wrestle, if he hadn’t felt it just then. The cold stale breath.
Presumably inspired by the IIHC's success, the Faux Faulkner award was created in 1989. (Both the Hemingway and Faulkner awards were later sponsored by United Airlines, which ended its sponsorship of both in 2005.) Perhaps the most prestigious extant bad-writing award is the Bulwer-Lytton fiction contest. Edward George Bulwer-Lytton was a Victorian novelist, whose novel Paul Clifford (1830) infamously begins with the line: It was a dark and stormy night... The Bulwer-Lytton, run by the English Department at San Jose State University, has been giving out their annual prize since 1982. Winners produce the worst first lines to novels from a variety of genres. The 2014 winner:
When the dead moose floated into view the famished crew cheered – this had to mean land! – but Captain Walgrove, flinty-eyed and clear headed thanks to the starvation cleanse in progress, gave fateful orders to remain on the original course and await the appearance of a second and confirming moose. — Elizabeth (Betsy) Dorfman, Bainbridge Island, WA
Or one of my favorites from this year, the winner of the purple prose category:
Cole kissed Anastasia, not in a lingering manner as a connoisseur might sip a glass of '82 La Pin, but open-mouthed and desperate, like a hobo wrapping his mouth around a bottle of Strawberry Ripple in the alley behind the 7-11. -- Terri Meeker, Nixa, MO
Typically, bad-writing contests seem to be named 'in honor of' specific writers, and take on the tone of good-natured joking rather than mockery. While the Bulwer-Lytton is the only major annual bad-writing contest I know of that's still giving out prizes, smaller once-off contests pop up frequently. Enjoy the following video, in which Neil Gaiman reads the winners of a special Bad-Gaiman contest.