For time-fractional parabolic equations with a Caputo time derivative of order alpha is an element of (0, 1), we give pointwise-in-time a posteriori error bounds in the spatial L2 and L infinity norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates 2 - alpha in the presence of solution singularities. (c) 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).