We concentrate on the parallel, fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetry-preserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions. In this article, motivated by [M. J. Gander, S. Loisel, D. B. Szyld, SIAM J. Matrix Anal. Appl. 33 (2012) 653–680] and [S. Nardean, M. Ferronato, A. S. Abushaikha, J. Comput. Phys. 442 (2021) 110513], we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain, and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems. Robustness, computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.