Communicating automata (CA) are a fundamental model of systems where a fixed finite number of processes communicate via message exchange through FIFO channels. In this paper, we introduce a parameterized version of CA (PCA). The parameter is the underlying communication topology, in which processes are linked via interfaces and arranged as graphs of bounded degree such as ranked trees, grids, rings, or pipelines. A given PCA can be run on any such topology. We provide Büchi-Elgot-Trakhtenbrot theorems for PCA, continuing the logical study that has established characterizations of classical CA in terms of (fragments of) monadic second-order (MSO) logic. In particular, we give translations of existential MSO logic to PCA that are correct for large and natural classes of topologies. Our main result relies on a locality theorem for first-order logic due to Schwentick and Barthelmann, and it uses, as a black-box, a construction by Genest, Kuske, and Muscholl from the non-parameterized case.