The increasing size and complexity of modern power systems and the integration of volatile renewable energy bring great challenges to the existing security-constrained unit commitment (SCUC) solution engines. This paper presents a fully parallel stochastic SCUC approach to obtain an efficient and fast solution for a large-scale power system with wind energy uncertainty. Variables duplication and auxiliary problem principle (APP) techniques are adopted to fully decompose the original stochastic optimization problem into three major solution modules: the unit commitment (UC) module solves multiple single UC problems; the optimal power flow (OPF) module handles multiple hourly DC-OPF problems; and the bridge module builds a connection between the UC and OPF modules. These three modules are conducted for both base case and scenarios, and can be totally solved in a parallel manner. Numerical case studies on a modified IEEE 118-bus system and a practical 1168-bus system demonstrate the effectiveness and efficiency of the proposed approach which will offer the power system a secure and economic operation under various uncertainties.