The story as far as I can tell is that 戲仿 is the original Chinese term for this sort of literary game.
The practice was not that common, but it was not unheard of either. The mid-Qing novel 鏡花緣 has a whole chapter describing games involving complicated parodies (chapter 87: 因舊事游戲仿楚詞 即美景詼諧編月令). The games involved a lot of drinking, and I had a lot of trouble following.
Words such as 諧仿 (or 諧擬, another candidate) are all later inventions, mostly the creations of academics who are trying to translate Western literary history, concepts, and theory into Chinese. Chinese Wikipedia is now 戲仿, after a struggle with 諧仿 some years ago.
I prefer 戲仿, since I'm interested in Chinese literature, and 戲仿 actually was used in poem titles by some poets.
PS: this is the best known parody in Taiwan (there are many, many versions, only the last line is always the same:
窗外蟬聲叫 Outside the windows the cicada sings
疑似下課鐘 I thought it was the end of class bell
抬頭看黑板 I lift up my head and look at the blackboard
低頭思便當 I lower my head and think of my lunchbox
(lunchboxes are round and made of stainless steel)