<p>In this paper, we investigate the optimal control problem governed by parabolic PDEs in random cylindrical domains, where the random domains are independent of time. We introduce a random mapping to transform the original problem in the random domain into the stochastic problem in the reference domain. The randomness of the transformed problem is reflected in the random coefficient matrix of the elliptic operator, the random time-derivative term, and the random forcing term. We make the finite-dimensional noise assumption on the random mapping in order to represent the random source of the transformed problem. Then, we use the perturbation method to expand the random functions in the transformed problem and establish the decoupled first-order and second-order optimality systems. Further, we combine the finite element method and the backward Euler scheme to obtain the fully discrete schemes for these two systems. Finally, the error analyses are respectively performed for the first-order and second-order schemes, and some examples are provided to verify the theoretical results.</p>