The past decade has seen clock-rate of individual processors stagnating, while parallel computational resources continue to increase rapidly. Spatial parallelization for the numerical solution of partial differential equations (PDEs) is well established, however such algorithms are reaching their strong scaling limit and further improvements to wall-clock time have necessitated the paradigm shift towards employing time-parallel algorithms in addition to space parallelization. Parallel-in-time methods have been successful in achieving speedup when computing solutions for a large number of PDEs. In this talk, we construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete Parareal iterations and incomplete iterations of the domain decomposition solver.