In recent years, I’ve noticed these words work their way into the mainstream vocabulary. Maybe I am wrong, but it seems their a fairly new to the mainstream American lexicon.
Are they legal terms?
No, but they are specialized.
Vet: to subject to a thorough examination. Originally, to subject to a thorough medical examination – from veterinary, “relating to animals or animal care” (or “beasts of burden” in its archaic sense). No, I don’t know how it came to be used this way, but I think it was originally a Britishism.
Parse: (of a sentence or phrase) to separate into or name all of the parts of speech. From pars, “part.”
redact: to edit; to select or adapt for publication; commonly, to censor for reasons of confidentiality. From redactus, PP of redigere, “to lead back; to bring back; to reduce.”
Almost definitely a bit of British English. Using “vet” as a verb in connection with animals seems to start around the 1890’s. It then, in the same decade, gets used about medically examining a person, and in 1904 Kipling uses it in the sense that you are asking about.
The use of vet as a simple examination is pretty clearly a shortening of veterinary combined with the penchant among Brits and Yanks to turn their nouns into verbs.
The specific extension of vet to mean an entrance exam or a winnowing process probably comes from the practice of vetting animals before races (to ensure that they are healthy enough to participate) or before shows (to reduce the possibility of spreading diseases among livestock when they are gathered).
For fairs and livewstock shows, a vet(erinary doctor) is generally required to examine each animal before it can be brought into the common pens. This reduces the chance that the Smith cattle that have been off on their own pastures with their own illnesses all summer will suddenly infect the cattle of the Fabers, the Schmidts, and the Ferronniers when they show up at the county fair.
For races, a similar practice prevents the running of a sick (or, now, a doped) horse. (I don’t know how prevalent doping was when the word came into currency to the point where it was cited in the OED in 1891.)