A stochastic model for theme park ride waiting times is developed by modeling the waiting time as a continuous-time, discrete-state Markov process with state-dependent, time-varying transition rates. These transition rates are interpreted as a control term acting on the waiting-time process, allowing the model calibration task to be formulated as a data-driven optimal control problem. To solve this problem efficiently, we construct a physics-informed neural network (PINN) that embeds the Kolmogorov forward equation solver into its architecture. Under mild assumptions, we prove the existence of an optimal control, providing theoretical support for the learning procedure. Numerical simulations demonstrate the effectiveness of the PINN-based solution. The proposed framework provides an interpretable, physically consistent, and data-driven approach for modeling and forecasting ride waiting times.