Crossing the boundaries of the space and time, and taking a word's journey across nations, cultural, political, and socio-economic worlds, is like chasing a wild goose.
Still, if one draws some parallels in sounds and follows well known rules of a classical language like Sanskrit, a well known ancient language in the Proto Indo European family of Languages, and perhaps the root itself, and adjusts sounds following well known grammatical and observed rules, to come to a known, related, or even an opposite meaning of the word, e.g. your profit is someones loss, then one is not necessarily far from the path the word took its so called traversal.
Parokaryote - Organisms not having cell nucleus. And they do not generate energy for living by burning oxygen, or doing aerobic respiration but by fermentation, or anaerobic respiration.
Parokaryote - (1) Pra + a + Kara + yata + i or (2) Para +Aakaara + yata + i
Paro -> Pro - as one expects constants to have a build in sounds. Like Ram and Rama, Atam and Atama.
Pro -> Prau -> Pra - Many people mix vowels - and a lot of cultures have a habit of saying vowels rounded to "o". Then there is a rule which says when a non "a" vowel combines with "e" -(ai), "o" -(au), - "ae" - (aai), and "ao" - (aau), one can drop the second vowels.
Pra -> Prefix for notion of going across forth, specially in verbs.
Akar -> A-Kar - > Action-less,
Para - Prefix for beyond.
Aakaar -> Form, Shape.
Para + Aa -> Paro - One of the rules that make "a + Aa" sound into "o" sound with encilitic. "Aa" comes from the word "Aakaar".
Aakaar -> Form, Shape.
yote -> yate - > yat -> which. Not sure why e is there, but it could mean masculine dual or feminine plural or even locative in ancient context.
So the meaning is that in "one which comes across forth with no action" or "one that has has gone beyond a shape", both of which comes close to the meaning of the word.
Eukaryote -> Aa-kaar-yat-e, Eka - kaar-yat-e, Aa-AaKaar -yate-e or Eka-Aakaar-yat-e--> Which has form or action. It means a living organism with cells having a nuclues or where DNA is bounded by a shape or form, and it has Aerobic metabolism - action of oxidation.
Now interesting, Parokaryotes have Anerobic Cellular Respiration, which means one without the oxygen, where fermentation is used to make energy, instead of oxidation. Oxidation is the most natural "action" of metabolism - known to human beings - meaning breathing oxygen!
Aerobic. So how about this word? Long story and so have patience.
Aries - Head of Horse or one of the Zodiacs. It is close to Airy in sound.
Ashvins meaning horses, the Twin Gods who bring Life to Dead, Physicians of the Heavens. They are also connected with Air, or Oxygen, which is Life Giving. Interestingly, Maruti or Maruts, the God of Winds, Rain and Thunder, like Indra or Zeus, as well as Ashvins are depicted with riding (seven) horses.
Aer - A + ir -> Aa + ir -> Aa + I + R -> Towards + Move + Prefix for Agent (like KartR from Karta) -> meaning something that moves towards - and that is Air.
Ira or iira-> Wind
iraa -> food, refreshment, enjoyment, draught or draft - like draft of wind or air.
Rob -> Rambh as well as Ram and Rabh(ah/as)-> Meaning embrace, take hold of, grasp, clasp, to act rashly, or to desire vehemently. Note Rob means close to Rambh. It is also connected to word "Grab" in English and "Grabh" in Sanskrit - which mean the same.
So Aerobic -> A + I + R + rambh + ic (prefix for Adjective) -> Aa + ira + rabh + ic -> A + e + Rob + Ic (dropping double "r" as one of the rules, removing aspiration of 'h', and making "a" as "o"), which becomes "Aerobic", meaning "embracing air or oxygen".
To make an opposite, add "an" as prefix (since the word is beginning with a vowel). So it becomes "an-aerobic", which is not with oxygen!
An-aerobic - One without oxygen.