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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