Why are parabolic subgroups called "parabolic subgroups"?

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.

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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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.