The large-scale integration of distributed renewable energy poses severe challenges to the secure and economic operation of distribution systems. In this paper, a novel nonparametric probabilistic optimal power flow (POPF) method of renewable distribution systems based on the Gaussian mixture model and Karush-Kuhn-Tucker conditions is proposed to evaluate the influence of renewable energy uncertainty on the distribution systems. The optimal power flow problem is solved with Karush-Kuhn-Tucker (KKT) conditions to obtain the analytical affine mapping relationship from renewable energy generation to the optimal solutions of output random variables, which significantly simplifies the optimization procedures. Then, the multivariate Gaussian mixture model is utilized to accurately describe the probabilistic characteristics of the correlated renewable energy. Finally, comprehensive numerical experiments on IEEE 33-bus test system verify the effectiveness of the proposed nonparametric POPF method for the distribution systems.