Why are parabolic subgroups called "parabolic subgroups"? 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?
 A: 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:

...There was no nice terminology for the subgroups $P _I$ with lie algebra the $\mathfrak p _I$ until R. Godement suggested calling them parabolic subgroups. I shall therefore anachronistically call them that...

"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.
A: 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 (...)
$\hspace{9em}$

A: My (completely non historical) point of view is the following. When you study non-compact symmetric spaces, e.g. the real hyperbolic space, isometries can be divided into three classes: elliptic (fixing a point in the space, so that it generates a relatively compact subgroup), hyperbolic (translates a geodesic, and acts like a dilation on the boudary of the space), and parabolic (none of the preceding type, but can be approximated both by elliptic and hyperbolic elements; always fixes a point on the boundary). In this context, a parabolic subgroup is the stabilizer of a point of the boundary, and contains many parabolic elements.
I guess that in a more algebraic (or should I say less geometrical?) context, this notion might generalize naturally to what is actually called a parabolic subgroup.
I hope this at least clarifies what is often meant by your answer #1.
