We introduce distributed event-triggered networked control of parabolic systems governed by semilinear diffusion PDEs. Sampled in time spatially distributed (either point or averaged) measurements are transmitted through a communication network to the controller only if a triggering condition is violated. We take into account quantization of the transmitted measurements and network-induced delays that are allowed to be larger than sampling intervals. We show that decentralized event-triggering mechanism can significantly reduce amount of transmitted measurements while preserving the system performance. (C) 2016 Elsevier Ltd. All rights reserved.