As stated in the title; I'm having a difficult time thinking of the equivalent conjugation. The word "Satire" is to "Satirically" as "Parody" is to ...?
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"Sardonically" could carries a connotation of disdain that is not always present in parody, and also doesn't necessarily imply imitation. What about "mocking" or "mockingly"?
I'm not sure if it's an official word, but parodiably might be correct. This is based on the related forms section of the definition here: http://dictionary.reference.com/browse/Parody
Also further down there is the word parodical, so parodically is a valid construction.
Also a parody can can be a satire, so sometimes satirically is the word you want.