This article investigates the event-triggered distributed model predictive control (DMPC) for perturbed coupled nonlinear systems subject to state and control input constraints. A novel compound event-triggered DMPC strategy, including a compound triggering condition and a new constraint tightening approach, is developed. In this event-triggered strategy, two stability-related conditions are checked in a parallel manner, which relaxes the requirement of the decrease of the Lyapunov function. An open-loop prediction scheme to avoid periodic transmission is designed for the states in the terminal set. As a result, the number of triggering and transmission instants can be reduced significantly. Furthermore, the proposed constraint tightening approach solves the problem of the state constraint satisfaction, which is quite challenging due to the external disturbances and the mutual influences caused by dynamical coupling. Simulations are conducted at last to validate the effectiveness of the proposed algorithm.