This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the dynamical-system state are available; these bases can be computed from snapshot data by proper orthogonal decomposition (POD) in the case of parameterized ODEs, for example. The method adopts the parareal framework for time parallelism, which is defined by an initialization method, a coarse propagator that advances solutions on a coarse time grid, and a fine propagator that operates on an underlying fine time grid. Rather than employing usual approaches for initialization and coarse propagation (e.g., a typical time integrator applied with a large time step), we propose novel datadriven techniques that leverage the available time-evolution bases. The coarse propagator is defined by a forecast (proposed in Ref.[12]) applied locally within each coarse time interval, which comprises the following steps: (1) apply the fine propagator for a small number of time steps, (2) approximate the state over the entire coarse time interval using gappy POD with the local time-evolution bases, and (3) select the approximation at the end of the time interval as the propagated state. We also propose both local-forecast initialization (i.e., the local-forecast coarse propagator applied sequentially) and global-forecast initialization (i.e., the local-forecast coarse propagator applied over the entire time interval with global time-evolution bases). The method is particularly well suited for POD-based reduced-order models (ROMs). In this case, spatial parallelism quickly saturates, as the ROM dynamical system is low dimensional; thus, time parallelism is needed to enable lower wall times. Further, the time-evolution bases can be extracted from readily available data, i.e., the right singular vectors arising during POD computation. In addition to performing analyses related to the method's accuracy, speedup, stability, and convergence, we also numerically demonstrate the method's performance. Here, numerical experiments on ROMs for a nonlinear convection-reaction problem demonstrate the method's ability to realize near-ideal speedups; global-forecast initialization with a local-forecast coarse propagator leads to the best performance.AMS subject classifications. 65B99, 65D30, 65L05A, 65L06, 65L20, 65M12, 65M55, 65Y05 1.1. Numerically solving ODEs: exposing concurrency. First, the sequential nature of numerically solving ODEs (i.e., numerical time integration) typically poses the dominant computational bottleneck, both in strong and weak scaling. Strong scaling refers to increasing the number of computing cores used to solve a problem of fixed (total) size. In the context of numerically solving ODEs, strong scaling is typically achieved through parallelizing 'across the system' by increasing the number of processors over which the problem is decomposed spatially; this usually associ...

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