Coupled mechanics and diffusion problems are very important in a variety of fields, e.g. in the modeling hydrogen diffusion-induced embrittlement in steels. State-of-the-art modeling of coupled multi-physics problems requires both physical sound, thermodynamically consistent material models and an efficient numerical framework to solve the systems of equations stemming from the finite element discretization of the underlying boundary value problem. Recently, a seamlessly integrated parallel solver framework, based on the Fast and Robust Overlapping Schwarz (FROSch) domain decomposition preconditioner [1] of the Trilinos software library (https://github.com/trilinos/trilinos) and on an incremental variational formulation of the coupled mechanics and diffusion problem, has been successfully applied in the modeling of hydrogels. Based on the implemented weak form of the corresponding minimization formulation, the influence of conforming and non-conforming finite element discretizations on the strong and weak scaling behavior has been studied in large-scale simulations [2] showing that conforming discretizations avoid unphysical results and are computationally more efficient. Within this contribution, we present scaling results of the saddle-point formulation of the coupled problem, report on the implications on the solver used and compare our findings to the results of the minimization formulation. Thus, the established modeling framework serves as a basis for further model extensions, incorporating e.g. phase field descriptions of Allen-Cahn or Cahn-Hilliard type as well as dissipative material models. [1] Heinlein A., Klawonn A., Rajamanickam S. and Rheinbach O. FROSch: A Fast and Robust Overlapping Schwarz Domain Decomposition Preconditioner based on Xpetra in Trillinos. In: Haynes R., et al. (eds.) Domain Decomposition Methods in Science and Engineering XXV, Lecture Notes in Computational Science and Engineering, Vol. 138, pp.176--184 (2020). DOI: 10.1007/978-3-030-56750-7_19 [2] Kiefer B., Prüger S., Rheinbach O. and Röver F. Monolithic parallel overlapping Schwarz methods in fully-coupled nonlinear chemo-mechanics problems. Computational Mechanics, 2023, https://doi.org/10.1007/s00466-022-02254-y.