Yes, parody is recognized as a type of fair use, like other commentary and criticism, and courts recognize that a parody must often take recognizable elements from the work it comments upon.
Courts do distinguish parody from satire. Parody copies from the object it mocks, while satire uses recognizable elements from the original work to mock something else or society in general. Parody gets broader fair use leeway than satire. If you want to make fun of Roy Orbison by changing “Pretty Woman” to “Big Hairy Woman,” that’s non-infringing parody; but if you make fun of the O.J. Simpson trial using Dr. Seuss illustrations and rhymes, that’s satire and in one famous case, it was found to be infringing (Campbell v. Acuff-Rose, Dr. Seuss Enterprises, L.P. v. Penguin Books U.S.A., Inc.).
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