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We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction to obtain symmetric matrices, which are typically better suited for iterative solvers. The MPI-parallel implementation uses the software libraries deal.II, p4est and FROSch (Fast of Robust Overlapping Schwarz). FROSch is part of the Trilinos library and is used in fully algebraic mode, i.e., the preconditioner is constructed from the monolithic system matrix without making explicit use of the problem structure. Strong and weak parallel scalability is studied using up to 512 cores, considering the standard GDSW (Generalized Dryja-Smith-Widlund) coarse space and the newer coarse space with reduced dimension. The FROSch solver is applicable to the coupled problems within in the range of processor cores considered here, although numerical scalablity cannot be expected (and is not observed) for the fully algebraic mode. In our strong scalability study, the average number of Krylov iterations per Newton iteration is higher by a factor of up to six compared to a linear elasticity problem. However, making mild use of the problem structure in the preconditioner, this number can be reduced to a factor of two and, importantly, also numerical scalability can then be achieved experimentally. Nevertheless, the fully algebraic mode is still preferable since a faster time to solution is achieved.