The optimal power flow (OPF) problem, a fundamental problem in power systems, is generally nonconvex and computationally challenging for networks with an increasing number of smart devices and real-time control requirements. In this paper, we first investigate a fully distributed approach by means of the augmented Lagrangian and proximal alternating minimization method to solve the nonconvex OPF problem with a convergence guarantee. Given time-critical requirements, we then extend the algorithm to a distributed parametric tracking scheme with practical warm-starting and termination strategies, which aims to provide a closed-loop sub-optimal control policy while taking into account the grid information updated at the time of decision making. The effectiveness of the proposed algorithm for real-time nonconvex OPF problems is demonstrated in numerical simulations.