The terminology is still not clear to me. What is the "unipotent radical" of an abstract group?

The word "reductive" (with its vague connotation of complete reducibility) once filled a vacuum in terminology, but now creates confusion in less classical parts of Lie theory and related group theory. Unlike "simple" or "semisimple" the word doesn't suggest specific structure, while there is often no connection to complete reduciblity of representations. By now we are stuck with the word but need to define it carefully in each context.

@Timothy: You may be expecting more rationality in the choice of terminology than exists. It's usually hard to come up with just the right word (standard or invented), so people may rely on (1) bland choices like "normal", (2) words transplanted from their original context like "parabolic", (3) names of people (appropriate or not) somehow associated with the concept --- the invented term "K3 surface" is one variant. After a while it's too late to go back and rethink the choices, as Freudentahl-deVries tried to do using highly nonstandard terminology in their 1969 book Linear Lie Groups.

P.S. A late instance of "parabolic" in connection with the modular group occurs in a 1974 thesis at NYU by the last student there of Wilhelm Magnus: Nonparabolic Subgroups of the Modular Group by Carol Tretkoff. But in line with Benoit's answer, the underlying rationale for the usage comes from study of homogeneous spaces such as $G/P$ in Lie theory. Borel himself didn't use the term "parabolic subgroup" in his 1956 Annals paper, but focused on complete/projective varieties starting with $G/B$. By 1962 he as well as Tits and others were using the term in print.

The invention of "parahoric" (after Iwahori) is apparently due to Bruhat-Tits in their follow-up work on structure theory over local fields following fundamental work by Iwahori and Matsumoto. Tits has always been fond of this kind of wordplay. (The introduction of "Borel subgroup" in his 1965 paper with Borel was probably due to Tits, though they left that ambiguous in a famous footnote.)

Borel's attribution of the terminology "parabolic subgroup" to Godement is reasonable, but Timothy Chow's first option probably comes closest to the rationale behind this choice. Study of the modular group by Fricke, Klein, and others distinguished several types of elements: "elliptic", "hyperbolic", "parabolic" (the latter typically coming from unipotent matrices). When Dan Mostow was asked about the origin of the naming convention back in 1977, I recall that he attributed it to the parallel with modular groups and parabolic elements. By 1962 Tits was using the term in his papers.

The answers and comments reinforce the importance of being precise about terminology used for orderings of Coxeter groups. Plus the widespread problem of having scattered computer programs that overlap but aren't readily found or used by other people. Such programs tend to grow out of someone's current research enthusiasm and then get put on the shelf or lack documentation or fail to run on some hardware. It's especially important to uniformize access to du Cloux's programs now that he is gone. But no one gets paid or rewarded for doing this extra work, while few are qualified.

The structure of finite groups has always been a rather specialized field ignored by most mathematicians. But the ordinary and modular representation theory has continued to have a fairly high profile, though here too it's somewhat of an acquired taste and usually not directly applicable elsewhere. George Lusztig has for example reshaped the agenda for groups of Lie type and attracted new interest in the area. Here the modular theory is still far from being understood. The classification of simple groups is another matter, having reached highest intensity around 1980.

@Qiaochu: Yes, I was thinking of Gian-Carlo Rota when I wrote that line. He actually wrote things down in a lot of places, having passionate views about invariant theory and its neglect. My own view of the subject has always been more tentative, but his personality is unforgettable for those of us who encountered him.

Yes, Hall was a major influence, especially in British group theory. I should emphasize that the first half of the 20th century, with its world wars, depression, and other upheavals, was not a flourishing period for mathematics. There were relatively few people doing real research, no "institutes" before IAS, disbanding of European research centers, few journals, slow communication. Even so, finite group theory, combinatorial group theory, Lie theory all made important progress. But it's true that finite group theory became less visible than other subjects in that period.

Surely somebody here will want to comment on the periodic rebirth of invariant theory?

This is probably overstated. The work of Frobenius, Schur, Burnside, Brauer, and others in the first decades of the 20th century took the subject in new directions. This grew out of 19th century invariant theory but opened the door (by the time of the Brauer-Fowler paper in 1955 on centralizers of involutions) to Feit-Thompson and the acceleration of the classification project. Representation theory of finite groups has kept developing in many directions, alongside the structure theory which levelled off more after 1980.

Small edit to last line of comment: "This many orbit closures at any rate ..."