We explore domain decomposition strategies for accelerating the convergence of all-at-once numerical schemes for the solution of time-dependent PDE-constrained optimization problems. The PDE constraints include the heat equation, Stokes equations, and advection-diffusion equation. All-at-once schemes aim to solve for all time-steps at the same time, which has the important advantages of preserving physical couplings in the solution, ensuring robustness with respect to the regularization parameter, and accelerating convergence. However, this approach leads to very large linear systems, with resulting costs in computation and memory. We describe possible parallel preconditioning strategies for these systems that use domain decomposition algorithms in the time domain and Schur complement approximations for the resulting local saddle point systems. We present numerical results showing the parallel performance of these algorithms and we discuss possible practical applications of the approach.