I've been thinking more about this. At first I thought, "well, Saturday Night Live makes money by creating parodies". However I did some more research this morning, and now stand behind my original post. For one big reason - I don't think a safety film, while using clear Star Trek references, actually parodies Star Trek. It's simply using ideas.
From PubLaw.Com, "The Supreme Court accepted 2 Live Crew's song as a parody because the rap song mimicked the original to achieve its message and because it "reasonably could be perceived as commenting on the original [Oh, Pretty Woman] or criticizing it, to some degree."
So, I guess the real question is this: does your video criticize or comment on Star Trek in some way? You could perhaps make the argument that you are commenting on the acting, cheeziness, the state of modern television, etc., but I think it would be a hard sell.
As for your bosses encouraging your work - bosses, I've found, are like most everyone else in the world. They think any material can be used in any way any time by anyone. Copyright law is a beast that most people rightly don't care about.