This paper proposes a Stochastic Mixed-Integer Linear Programming (SMILP) formulation for Short-Term Hydrothermal Generation Scheduling (STHTGS) under uncertainty. STHTGS seeks to minimize present and future operation costs by deciding the commitments of thermal generators and the allocation of hydro resources during the planning horizon. The stochastic STHTGS is decomposed using the Progressive Hedging Algorithm (PHA) and each sub-problem is solved in parallel. Numerical tests are conducted for the Chilean Central Interconnected System with 12 stochastic scenarios and a weekly decision horizon. The stochastic and deterministic formulations are compared by solving standard variations of the stochastic problem. Numerical results show that the proposed decomposition and parallelization strategy can help reduce simulation times and hedge against uncertainty, but the level of benefits and the convergence properties are highly dependent on the amount of water available in the scheduling horizon and the diversity of the scenarios.