Biological populations can be faced with two detriments simultaneously if they experience both parasitism and an Allee effect. While infection with disease causes additional mortality, the Allee effect is a demographic process describing depensation (i.e., population decline or reduced population growth at low densities in case of a 'strong' or 'weak' Allee effect, respectively). The joint interplay of disease spread and a strong Allee effect are investigated in mathematical models that consist of two differential equations (describing the susceptible and infectious part of the host population) with a cubic nonlinearity (modelling the Allee effect). Two different incidences are considered, namely frequency-and density-dependent transmission, which model the infection process at two opposite ends of a spectrum of possibilities. Various threshold quantities are derived and employed to explain infection disappearance, parasite invasion and host extinction. The comparison of dynamical behaviour in both models provides interesting insight how depensation and disease transmission interact at various population densities. The general impact of disease is (i) to depress the host population size in endemic equilibrium and (ii) to enlarge the likelihood of extinction. If the incidence is density-dependent, oscillatory dynamics are possible as well as the emergence of three endemic equilibria, rendering the population tristable. The latter scenario is discussed in detail with respect to implications for the conservation of endangered species and the management of pests such as invasive alien species. Critical parameter values are identified for which population persistence might be possible even at extremely large values of the basic reproduction number R_0, which could be expected to drive the host extinct independent of the initial condition.