This paper compares two common approaches to achieve robustness in the parallel multigrid resolution of anisotropic operators on structured grids: alternating plane smoothers with full coarsening and plane smoothers combined with semicoarsening. Both, numerical and architectural properties are compared. A parallel implementation based on MPI is studied in a realistic way by considering the exploitation of the cache memory and the negative effect that the parallelization has on the convergence properties of the methods.