Moral rights in Copyright
Maybe it's different in the USA, but part of copyright law in Europe (The Berne Convention for the Protection of Literary and Artistic Works (1886)) is the right to reputation:
"The Convention also protects moral rights, giving the author of a work the right to claim authorship of the work and the right to object to any mutilation or deformation or other modification or derogatory action to the work which would be prejudicial to the author’s honour or reputation."
It seems to me that a parody that represents itself as the actual work of the parodied party would overstep this aspect of the law. The more convincingly it does so, the more would infringe.
I am not a lawyer. This is not advice. You have not been charged.