Couplings in multi-physics problems add a facet of complexity on their mathematical analysis and numerical simulation [M. Anselmann, M. Bause, N. Margenberg, P. Shamko, An energy-efficient GMRES--Multigrid solver for space-time finite element computation of dynamic poroelasticity, Comput. Mech., 74 (2024), pp. 889--912; doi: 10.1007/s00466-024-02460-w], [J. S. Stokke, M. Bause, N. Margenberg, F. A. Radu, The Biot–Allard poro-elasticity system: equivalent forms and well-posedness, Appl. Math. Lett., 158 (2024), 109224 (6 pages); doi: 10.1016/j.aml.2024.109224]. A reason for this might be different mathematical characteristics of solutions to the subproblems involved in the overall systems, strong variations in parameters or degenerations. Here, we study the hyperbolic-parabolic system of dynamic poroelasticity. A natural and promising approach for the numerical approximation of coupled systems is obtained by space-time finite element methods (STFEMs). They offer the natural construction of higher order schemes that achieve accurate results on computationally feasible grids with a minimum of numerical costs. In particular, this applies to 2nd order problems of wave propagation. Here, we study families of STFEMs to the system of dynamic poroelasticity rewritten as a first-order system in time or, alternatively, as a first-order system in space and time [M. Bause, S. Franz, M. Anselmann, Structure preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system}, Electron. Trans. Numer. Anal., accepted (2024), 27 pages; arXiv:2311.01264]. Appreciable advantage of the latter approach is its structure preservation and the accurate stress and flux approximation as part of the formulation itself. Error estimates are discussed. Preconditioners tailored to the structure of the algebraic systems are presented. Their performance properties are carefully analyzed.