In this paper, accurate and fast computations are proposed for an original set of coupled three-dimensional spatial problems of anisotropic diffusion. A spatial-temporal filter is presented for dynamic positron emission tomography imaging in which diffusion is based on the distances between vectorial representations of the voxel intensity across time. Spatial discretization is proposed combined with explicit or semi-implicit discretization schemes in which the stability of the global scheme is studied. At each time step, it is necessary to solve m large linear coupled algebraic systems for which two levels of parallelism are considered. The first level is due to the fact that the m images can be processed independently once the coefficient of diffusion is computed. The second level of parallelism appears in the solution of each algebraic system, which can be solved by using parallel synchronous and asynchronous iterative methods. For the previous parallel iterative methods, we analyze the convergence. Experimental parallel simulations concerning 3D+t anisotropic diffusion of simulated dynamic positron emission tomography images are presented.