In this paper, we consider the distributed parameter estimation problem over sensor networks in the presence of quantized data and directed communication links. We propose a two-stage algorithm aiming at achieving the centralized sample mean estimate in a distributed manner. The running average technique is utilized in the proposed algorithm to smear out the randomness caused by the probabilistic quantization scheme. It is shown that the centralized estimate can be achieved in the mean square sense, which is not observed in the conventional consensus algorithms. Simulation results are presented to illustrate the effectiveness of the proposed algorithm and highlight the improvements by using running average technique.