We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The targeted application is the poroelasticity system within the quasi-static assumption. An abstract setting is proposed to identify a natural energy norm for the PDE system. Two a posteriori error analyzes are performed, both yielding reliable upper error bounds in the sense that all the constants are specified. The first analysis hinges directly on the stability of the continuous problem and can be used to estimate the dominant term associated with the p-component in the energy norm. The second analysis is an extension of the elliptic reconstruction technique introduced by Makridakis and Nochetto for linear parabolic problems. It is used here to derive an a posteriori error estimate for the u-component in the energy norm that exhibits an optimal convergence rate with respect to mesh size. Numerical results are presented to illustrate the performance of the various estimators.