Over the years, I have heard two different proposed answers to this question.
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It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a really convincing explanation along these lines.
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"Parabolic" is short for "para-Borelic," meaning "containing a Borel subgroup."
Which answer, if either, is correct?
A related question is who first introduced the term and when. Chevalley perhaps?
Best Answer
It appears that neither of the answers is fully correct. There is a great book, "Essays in the history of Lie groups and algebraic groups" by Armand Borel, when it comes to references of this type. To quote from chapter VI section 2:
"The geometry of the finite simple groups" by F. Buekenhout is on the other hand the only paper that came up in a search for paraborelic, and the author mentions he is using this term instead of parabolic to distinguish from parabolic subgroups of Chevalley groups.