Basically, as I understand it, the reason that parody is allowed is the following: It would be a violation of the first amendment to prevent criticism of writing or artwork. Parodying a work of art to make fun of that same piece of art falls into the category of critiquing it.

Acting/Opera companies (like the G&S Players) probably make minor modification with the approval of the licencing company, or perform pieces that have fallen into the public domain (G&S worked together in the late 1800s), or they hope not to be called on it.

Didn't know that about covers (which are certainly not parody). I know there was a major issue a few years ago about performers that make digital samples from several performers, and create "songs" that are aural mosaics. Copyright gets dicey at times.