In this paper we propose and describe a parallel implementation of a block preconditioner in the solution of saddle point linear systems arising from Finite Element (FE) discretization of 3D coupled consolidation problems. The inexact constraint preconditioner recently developed is combined with the parallel FSAI preconditioner which is used here to approximate both the structural (1,1) block and the Schur complement matrix. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. Numerical results on a number of test cases of size up to 2×106 unknowns and 1.2×108 nonzeros show the perfect scalability of the overall code up to 256 processors. The proposed iterative method is shown to outperform the well-established pARMS multilevel solver on the largest problem.