There is more leverage with Parody than with drama, but your admixture of original sound makes the issue of 'fair use' here complicated. However, the truth is, most folk just go forward and if they get a C&D (cease and desist) letter from CBS or Paramount, they don't test their rights, they do cease and desist. You probably have nothing to lose by posting it. Chances are nothing worse will happen than it will be taken down.
If you do post it, I'd like a head-up so I could add it to my website, which is an index of Star Trek Fan Films. Even if you don't post it, I'd like to have it's name so I can include it on my list of Star Trek Fan Films not available for viewing by the general public.