I disagree that evolutionary arms race is a specific term of art; we have many specific terms of art but 'arms race' is a broad generalization popularized by Dawkins as a pop science writer addressing a lay audience. Actual terms of art in this area would include Red Queen, the many individually termed coevolutions (antagonistic, mosaic, host-parasite, plant-herbivore, predator-prey etc), coadaptation, coextinction, the escalation hypothesis, frequency-dependent selection, reciprocal selection, asymmetric selection, the evolutionary lag, evolutionary cycling, character displacement, Fisherian runaway, evolutionary mismatch/trap, (phylogenetic) niche conservatism, fitness landscape, Grinnellian vs Eltonian niches, the competitive exclusion principle, and on and on. All of these actual terms of art fit under the broad, general umbrella of an 'arms race' with other species, which is really nothing more than a restatement of Spencer's unfortunate phrase. The latter is so widely 'known' that it is to the point that I and many of my peers try not to utter it, in an effort to reduce the work refuting the same tired misunderstandings that arise from that verbiage.
At any rate, almost NONE of these actual terms of art are about the sort of equilibrium that was the exact heart of the OP's query to the LLM, and thus nearly none of the broader umbrella 'arms race' is about why the plant doesn't have the evolutionary pressure to actually drive the parasite extinct. An arms race doesn't have to be in equilibrium. Armor vs weapons were in an arms race and indeed at equilibrium for millenia, but then bullets come along and armor goes exinct almost overnight and doesn't reappear for 5 centuries. Bullets win the arms race. Arms races have nothing to do, inherently, with equilibrium.
You seem to have misunderstood the nature of the equilibrium in a Red Queen scenario, which is the fundamental effect that the hypothesis is directly named for. That species that are in Red Queen relationships can go extinct is in no way a counterargument to the idea that two (or more) species tend to coevolve in such a way that the relative fitness of each (and of the system as a whole) stays constant. See, for example, the end of the first paragraph on the origin of Van Valen's term at your own wiki link.
Evolutionary steady-state is a synonymous term without the baggage of the literary reference and also avoids the incorrect connotation suggested by arms race that leads people to forget the abiotic factors that are often a dominant mechanism in extinctions as the realized niche vs the fundamental niche differ. Instead, Van Valen was specifically proposing the Red Queen hypothesis as an explanation of why extinction appears to be a half-life, i.e. of a constant probability, rather than a rate that depends on the lifetime of the taxa. This mechanism has good explanatory power for the strong and consistent evidence that speciation rate (usually considered as the log of the number of genera, depending on definition, see Stanley's Rule) has a direct and linear relation with the extinction rate. If Red Queened species didn't go exinct, Van Valen wouldn't have needed to coin the term to explain this correlation.
Or were you deliberately invoking Cunningham's Law?