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• A new algorithm incorporates model reduction in the spatial and temporal domains. • We study parabolic problems with heterogeneous coefficients and rough initial data. • We derive convergence analysis that weakly depends on the heterogeneous coefficients. • The convergence is rigorously studied, which greatly improves the current result. • Extensive numerical tests are performed to show the fast convergence of our algorithm. We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Extensive numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly. [ABSTRACT FROM AUTHOR]