With increasing grid modernization efforts, future electric grids will be governed by more complex and faster dynamics with the high penetration of new components such as power electronic-based control devices and large renewable resources. These lead to the importance of developing real-time dynamic security assessment under the consideration of uncertainties, whose main tool is time-domain simulation. Though there are many efforts to improve the computational performance of time-domain simulation, its focus has been on the deterministic differential-algebraic equations (DAEs) without modeling uncertainties inherent in power system networks. To this end, this paper investigates large-scale time-domain simulation including effects of stochastic perturbations and ways for its computational enhancement. Particularly, it utilizes the parallel-in-time (Parareal) algorithm, which has shown great potentials, to solve stochastic DAEs (SDAEs) efficiently. A general procedure to compute the numerical solution of SDAEs with the Parareal algorithm is described. Numerical case studies with 10-generator 39-bus system and 327-generator 2383-bus system are performed to demonstrate its feasibility and efficiency. We also discuss the feasibility of employing semi-analytical solution methods, using the Adomian decomposition method, to solve SDAEs. The proposed simulation framework provides a general solution scheme and has the potential for fast and large-scale stochastic power system dynamic simulations.