The question as worded implies that if something is a parody it is automatically fair use or allowed in US copyright law. This is a myth.
First of all, in a copyright context, the term "parody" is somewhat limited. In that sense, a "parody" is a new work which comments on the original (often but not always by mocking or ridiculing the original). A mere alternate version which modifies the form of the original, perhaps humorously, but not to comment on the original or perhaps on much of anything, is not a parody. A new work which modifies the original to comment on something else, but not on the original, is a satire. Of course many works may be both parodies and satires in this sense.
See this law.se Q&A for extensive quotes from Suntrust Bank v. Houghton Mifflin Co 268 F.3d 1257 (The case of The Wind Done Gone vs Gone with the Wind) and discussion of what is and is not a parody in US copyright law, and when a parody is fair use.
A parody, because it comments on the original, will often be found to be a fair use of the original. But not always. The full four-factor analysis must still be done, and the results are never certain until a court has passed on the specific case. See this law.se Q&A for more on fair use.
I can't tell from the question whether the modified song is truly a parody in the sense used in copyright law. If not, it probably isn't fair use, although it might be for other reasons.
By the way, a song written for a television show or video to be distributed commercially is a commercial use, even if a non-profit corporation is involved, and even if there is an educational purpose.
Note that fair-use is a strictly US legal concept. Even if something is fair use under US law, it may be copyright infringement under the laws of some other country.
