In this paper, we present a robust distributed model predictive control (DMPC) scheme for dynamically decoupled nonlinear systems which are subject to state constraints, coupled state constraints and input constraints. In the proposed control scheme, all subsystems solve their local optimization problem in parallel and neighbor-to-neighbor communication suffices. The approach relies on consistency constraints which define a neighborhood around each subsystem's reference trajectory where the state of the subsystem is guaranteed to stay in. Contrary to related approaches, the reference trajectories are improved consecutively. In order to ensure the controller's robustness against bounded uncertainties, we employ tubes. The presented approach can be considered as a time-efficient alternative to the well-established sequential DMPC. In the end, we briefly comment on an iterative extension. The effectiveness of the proposed DMPC scheme is demonstrated with simulations, and its performance is compared to other DMPC schemes.