Successful invasion of a parasite into a host population and resulting host-parasite dynamics can depend crucially on other members of a host's community such as predators. We do not fully understand how predation intensity and selectivity shape host-parasite dynamics because the interplay between predator density, predator foraging behavior, and ecosystem productivity remains incompletely explored. By modifying a standard susceptible-infected model, we show how productivity can modulate complex behavior induced by saturating and selective foraging behavior of predators in an otherwise stable host-parasite system. When predators strongly prefer parasitized hosts, the host-parasite system can oscillate, but predators can also create alternative stable states, Allee effects, and catastrophic extinction of parasites. In the latter three cases, parasites have difficulty invading and/or persisting in ecosystems. When predators are intermediately selective, these more complex behaviors become less important, but the host-parasite system can switch from stable to oscillating and then back to stable states along a gradient of predator control. Surprisingly, at higher productivity, predators that neutrally select or avoid parasitized hosts can catalyze extinction of both hosts and parasites. Thus, synergy between two enemies can end disastrously for the host. Such diverse outcomes underscore the crucial importance of the community and ecosystem context in which host-parasite interactions occur.