In the framework of the numerical solution of evolutionary differential problems, the limitations of sequential computing have prompted the need to design parallel methods. The aim of this paper is to bring together the parallelization in time of the parareal algorithm and that in space of splitting techniques, yielding a new type of space-time parallel methods. Their potential in the context of reaction-diffusion problems is illustrated by solving the Gray-Scott model, a well-known problem in the field of mathematical biology that describes the formation of animal patterns.