Note that the captain calls one female officer "Number One", so it also drew from "The Menagerie" ("The Cage" being unknown in Australia in the 60s). The "Mavis Bramston Show" ran 1964-68, so it was sending up "Star Trek" as a then-current show.
Here's the "Paul Hogan Show" (late 70s/early 80s) version. It once received a write-up in "Starlog" magazine, long before Hogan became a household name in the USA. The skit uses the same Spock-in-the-elevator joke, not just the title of the old MBS skit: