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Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for instance, arising in linear elasticity, the existing two-level domain decomposition solvers with the vertex-based coarse grid show poor numerical and parallel scalabilities. Therefore, new algorithms with a better resolved coarse grid are needed. The probabilistic wirebasket-based coarse grid for a two-level solver is devised in three dimensions. This enriched coarse grid provides an efficient mechanism for global error propagation and thus improves the convergence. This development enhances the scalability of the two-level solver in handling stochastic PDEs in three dimensions. Numerical and parallel scalabilities of this algorithm are studied using MPI and PETSc libraries on high-performance computing (HPC) systems. Implementational challenges of the intrusive spectral stochastic finite element methods (SSFEM) are addressed by coupling domain decomposition solvers with FEniCS general purpose finite element package. This work generalizes the applications of intrusive SSFEM to tackle a variety of stochastic PDEs and emphasize the usefulness of the domain decomposition-based solvers and HPC for uncertainty quantification.