Parody and paradox, despite superficial similarities in pronunciation, are unrelated concepts.
In literature, parodies are literary works that imitate other works of literature for humorous purposes, often to denigrate what the author of the parody considers trite or absurd features of the original being parodied. For example, Anthony Hecht's "Dover Bitch" parodies Matthew Arnold's "Dover Beach" by using the viewpoint of the wopman being addressed to make fun of the sentimentality of Arnold's work. Swinburne's "Heptalogia" parodies what he sees as the triteness of many Victorian poets (and includes a delightful self-parody). The mock-epic (e.g. "Orlando Furioso" and "Rape of the Lock") parodies the grandeur of epic by injecting into the form either a trivial subject (stealing a lock of hair as opposed to the "Rape of Persephone, in which the daughter of a goddess is abducted by Hades) or a trivial approach to a serious subject (in Orlando Furioso, the knight, blinded by their helms and clumsy in their armor, do a great deal of damage to the surrounding scenery but little to each other in a duel). The Brand-X Anthology of Poetry contains many examples of parody.
Paradox is a philosophical concept of an inherently irresolvable self-contradiction that leads one to revisit the plausibility of certain premises for philosophical thought (see Quine, "Ways of Paradox"). Kant's "Antimonies of Pure Reason" and Abelard's "Sic et Non" are examples. Zeno's famous paradoxes (that it is impossible to cross a room, becuase first one would need to cross half the distance, then half the remaining distance, etc. ad infinitum) were intended to refute the physical theories of the pluralists by showing the absurdity of their assumptions (and were not, in fact, fully resolved until the twentieth century discovery of space being quantized and thus not infinitely subdivisiable). The paradox is related to the "reductio ad absurdum." In literature, the common ancient sophistic genre of "paradoxical encomium" praised something inherently unpraiseworthy, such as the "Encomium of a Flea" which influenced John Donne's poem "The Flea".