This research develops a hybrid stochastic-robust approach for the optimal operation of integrated urban gas and electricity distribution networks (IUEAGDNs) based on convex optimization under the high penetration of renewable energy sources and parking lots. The aim of this study is to minimize operating costs and increase the flexibility of the IUEAGDNs by coordinating the smart charging strategy of electric vehicles (EVs) and the natural gas storage system under uncertainty conditions. The structure of smart parking lots is modeled using the k-mean clustering method, taking into account the arrival time, departure time, and miles traveled by EVs. The uncertain behavior of EVs and electric and natural gas loads has been modeled using a stochastic optimization method. Meanwhile, fluctuations in wholesale electricity market prices have been addressed through a robust optimization method. Additionally, this study explores the impact of integrating gas storage systems into the natural gas network, focusing on enhancing flexibility and improving fuel supply to gas-fired distributed generation (GFDG) sources. The proposed model is a second-order conic programming (SOCP) that guarantees a global optimal solution to this problem. The obtained results strongly confirm the effectiveness of the proposed structure.