Ensuring reliable operation in power grids given the complexity of modern electricity networks with increasing demands, varying generation and flows present significant computational challenges. Addressing these challenges, we develop efficient methods for solving multi-period optimal power flow (MPOPF) problems. The class of MPOPF problems is NP-hard due to the non-convexity of power flow equations and due to the temporal coupling of the decision variables introduced by ramp constraints. To comprehensively address these challenges, we extend a conic relaxation, originally developed for single-period optimal power flow problems, to convexify the MPOPF problem, and develop a customized alternating direction method of multipliers to solve it in parallel. The proposed method is guaranteed to converge and provides a tighter lower bound than standard second-order cone relaxations. We assess the performance of our method on standard systems.