To answer this question, Spencer R. Hall, Meghan A. Duffy, and Carla E. Cáceres studied a simple model which shows how predators that strongly prefer parasitized hosts can introduce "Allee effects" for parasites at lower productivity and sudden but unavoidable extinction of parasites at higher productivity. In the former case, parasites have difficulty invading a host population, while in the latter case, parasites cannot persist with their hosts, even if they can invade. When predators less strongly prefer parasitized hosts, these dramatic behaviors diminish, but predators can still trigger large fluctuations of parasite abundance at higher productivity.
Surprisingly, when predators show no preference for or even avoid parasitized hosts, interactions between the two enemies can drive both host and parasite unavoidably extinct. These findings stress the importance of the biological and ecosystem background in which hosts and parasites interact.