Generative adversarial networks (GANs) are widely used to learn generative models. GANs consist of two networks, a generator and a discriminator, that apply adversarial learning to optimize their parameters. This article presents a parallel/distributed implementation of a cellular competitive coevolutionary method to train two populations of GANs. A distributed memory parallel implementation is proposed for execution in high performance/supercomputing centers. Efficient results are reported on addressing the generation of handwritten digits (MNIST dataset samples). Moreover, the proposed implementation is able to reduce the training times and scale properly when considering different grid sizes for training.