Galaxy clusters are the largest objects in the universe that are held together by their own gravity. Most of their baryonic content is made of a magnetized diffuse plasma. We investigate the impact of such a magnetized environment on ultra-high-energy cosmic-ray (UHECR) propagation. The intracluster medium is described according to the self-similar assumption, in which the gas-density and pressure profiles are fully determined by the mass and redshift of the cluster. The magnetic field is scaled to the thermal components of the intracluster medium under different assumptions. We model the interactive and diffusive processes of UHECRs in the intercluster medium using a modified version of the Monte Carlo code {\it SimProp}, where hadronic processes and magnetic confinement are implemented, as well as the effect of the cluster's magnetic field. We provide a universal parametrization of the UHECR flux escaping from the environment as a function of the most relevant quantities, such as the mass of the cluster, the position of the source with respect to the center of the cluster and the nature of the accelerated particles. We show that galaxy clusters are an opaque environment, especially for UHECR nuclei. The role of the most massive nearby clusters in the context of the emerging UHECR astronomy is finally discussed.