TITLE: Why are parabolic subgroups called "parabolic subgroups"?
QUESTION [51 upvotes]: Over the years, I have heard two different proposed answers to this question.

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.
"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?

REPLY [15 votes]: The naming (attributed by Borel as quoted by @GjergjiZaimi) happened quite publicly, in Roger Godement : Groupes linéaires algébriques sur un corps parfait, Sémin. Bourbaki 13 (1960/61), Exp. No. 206, 22 p. (1961). ZBL0119.27206, first page (my bold & link):

When $G_A/G_{\mathbf Q}$ is not compact, it is equally easy to conjecture that one must be able to define something like Poincaré’s classical “parabolic cusps”, which must correspond to nontrivial unipotent subgroups of $G_{\mathbf Q}$ (...) We shall, in this talk, define and study “parabolic subgroups” by methods of algebraic group theory.

Godement describes this at length in his Analyse mathématique IV, e.g. p. 441: in the Poincaré upper half-plane,

a parabolic cusp with vertex $\xi$ is the part of a horocycle with center $\xi$ comprised between two arcs of circle orthogonal to the real axis at $\xi$; see the figure (...)
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