The classical approach for solving evolution Partial Differential Equations (PDEs) using a parallel computer consists in first partitioning the spatial domain and assigning each subdomain to a processor to achieve space-parallelism, then advancing the solution sequentially. However, enabling parallelism along the time dimension, despite its intrinsic difficulty, can be of paramount importance to fast computations when space-parallelism is unfeasible, cannot fully exploit a massively parallel machine or when near-real-time prediction is desired. The aforementioned objective can be achieved by applying classical domain decomposition principles to the time axis. The latter is first partitioned into time-slices to be processed independently. Starting with approximate seed information that provides a set of initial conditions, the response is then advanced in parallel in each time-slice using a standard time-stepping integrator. This decomposed solution exhibits discontinuities or jumps at the time-slice boundaries if the initial guess is not accurate. Applying a Newton-like approach to the time-dependent system, a correction function is then computed to improve the accuracy of the seed values and the process is repeated until convergence is reached. Methods based on the above concept have been successfully applied to various problems but none was found to be competitive for even for the simplest of second-order hyperbolic PDEs, a class of equations that covers the field of structural dynamics among others. To overcome this difficulty, a key idea is to improve the sequential propagator used for correcting the seed values, observing that the original evolution problem and the derived corrective one are closely related. The present work first demonstrates how this insight can be brought to fruition in the context of linear oscillators, with numerical examples featuring structural models ranging from academic to more challenging large-scale ones. An extension of this method to nonlinear equations is then developed and its concrete application to geometrically nonlinear transient dynamics is presented. Finally, it is shown how the time-reversibility property that characterizes some of the above problems can be exploited to develop a new framework that provides an increased speed-up factor.