The naming (attributed by Borel as quoted by @GjergjiZaimi) happened quite publicly, in <cite authors="Godement, Roger">_Roger Godement_ : [**Groupes linéaires algébriques sur un corps parfait**](http://www.numdam.org/item?id=SB_1960-1961__6__11_0), Sémin. Bourbaki 13 (1960/61), Exp. No. 206, 22 p. (1961). [ZBL0119.27206](https://zbmath.org/?q=an:0119.27206)</cite>, 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](https://fr.wikipedia.org/wiki/Courbe_modulaire)”**, 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](https://zbmath.org/?q=an:1030.28001), 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 (...)

>$\hspace{9em}$<img src="https://i.sstatic.net/jvxqz.png" width="278" height="164">