In short, parodies can be in compliance with copyright law, but it would ultimately depend on what your final product looks like.
The often cited (and sometimes misunderstood) fair use section of copyright law is one of several limitations on exclusive ownership. The entire section is as follows:
Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include—
(1)the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2)the nature of the copyrighted work;
(3)the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4)the effect of the use upon the potential market for or value of the copyrighted work.
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors.
Over time, high courts have agreed that parodies can be considered a genre of criticism and commentary. Here is a list of major court cases where the ultimate issue was centered around what is and is not fair use.
Legally speaking, you may produce a parody of another copyrighted work without running afoul with civil law. If you read the Fair Use Index linked above, however, you may notice that other people/organizations have tried unsuccessfully to claim fair use as a defense for their work. Their stories should serve as a reminder that while fair use generally allows parodies, it does not grant you a license to do whatever you want with another owner's copyrighted work if your argument of commentary or criticism is weak.
As for practicality, there is always the possibility that Viacom, Disney, or whoever owns South Park Studios nowadays can make legal threats and scare smaller platforms like Newgrounds into removing the content despite the merits of your fair use claim.
I would say that because parodies tend to be a copyright safe zone for animators, along with the fact that Newgrounds itself proudly hosts an extensive catalogue of parodies dating back a quarter of century, you are probably okay to submit it so long as it passes the Under Judgment process and has enough artistic merit to be distinguished from the original work.