This paper studies an averaged Linear Quadratic Regulator (LQR) problem for a parabolic partial differential equation (PDE), where the system dynamics are affected by uncertain parameters. Instead of assuming a deterministic operator, we model the uncertainty using a probability distribution over a set of possible system dynamics. This approach extends classical optimal control theory by incorporating an averaging framework to account for parameter uncertainty. We establish the existence and uniqueness of the optimal control solution and analyze its convergence as the probability distribution governing the system parameters changes. These results provide a rigorous foundation for solving optimal control problems in the presence of parameter uncertainty. Our findings lay the groundwork for further studies on optimal control in dynamic systems with uncertainty.