The computational kernel of scientific codes is often a linear system solver. Krylov subspace methods may lead to scalable linear system solvers when provided with robust and efficient preconditioners. We describe two parallel domain decomposition preconditioners for linear systems of equations arising from the piecewise Hermite bicubic collocation discretization of second-order elliptic PDEs defined on a rectangular domain and with general boundary conditions. The preconditioners use an original three-level discretization scheme derived from partitioning the domain into subdomains, edges, and vertices. We describe the parallel implementation and evaluate the robustness and efficiency of the preconditioners, using numerical experiments. We outline the advantages and disadvantages of selecting an OpenMP or MPI implementation, and compare the performance of our approach with that of a PETSc implementation.