Future biodiversity loss threatens the integrity of complex ecological associations, including among hosts and parasites. Almost half of primate species are threatened with extinction, and the loss of threatened hosts could negatively impact parasite associations and ecosystem functions. If endangered hosts are highly connected in host-parasite networks, then future host extinctions will also drive parasite extinctions, destabilizing ecological networks. If threatened hosts are not highly connected, however, then network structure should not be greatly affected by the loss of threatened hosts. Networks with high connectance, modularity, nestedness and robustness are more resilient to perturbations such as the loss of interactions than sparse, nonmodular and non-nested networks. We analysed the interaction network involving 213 primates and 763 parasites and removed threatened primates (114 species) to simulate the effects of extinction. Our analyses revealed that connections to 23% of primate parasites (176 species) may be lost if threatened primates go extinct. In addition, measures of network structure were affected, but in varying ways because threatened hosts have fewer parasite interactions than non-threatened hosts. These results reveal that host extinctions will perturb the host-parasite network and potentially lead to secondary extinctions of parasites. The ecological consequences of these extinctions remain unclear. This article is part of the theme issue 'Infectious disease macroecology: parasite diversity and dynamics across the globe'.