The surrogate-based POSEIDON algorithm for high dimensional discrete multi-objective black-box optimization with constraints has shown good performance on the Mazda benchmark car structure design problem that has 222 discrete variables and 54 black-box inequality constraints. This paper explores the possibility of improving the wall clock time of POSEIDON to find an approximate Pareto front through parallel processing. The algorithm is modified to allow the selection of multiple points for simultaneous objective and constraint function evaluations in parallel. Within a given parallel iteration, the selection of points for function evaluations is done sequentially, using not only the information from previously evaluated sample points and also previously chosen points within the current iteration. Numerical experiments on the Mazda benchmark demonstrate that a parallel implementation of POSEIDON using up to 8 processors provides considerable improvement over the serial version on the set of nondominated points obtained as measured by the hypervolume metric based on a given reference point at various computational budgets. However, the improvements do not seem to scale very well as the number of processors increase on the Mazda benchmark. Nevertheless the parallel algorithm considered represents some progress that may be used as a stepping stone for designing a more scalable parallel algorithm for constrained discrete multi-objective black-box optimization.