In this paper, we propose an efficient iterative method called RB-iteration, based on reduced basis (RB) techniques, for addressing time-dependent problems with random input parameters. This method reformulates the original model such that the left-hand side is parameter-independent, while the right-hand side remains parameter-dependent, facilitating the application of fixed-point iteration for solving the system. High-fidelity simulations for time-dependent problems often demand considerable computational resources, rendering them impractical for many applications. RB-iteration enhances computational efficiency by executing iterations in a reduced order space. This approach results in significant reductions in computational costs. We conduct a rigorous convergence analysis and present detailed numerical experiments for the RB-iteration method. Our results clearly demonstrate that RB-iteration achieves superior efficiency compared to the direct fixed-point iteration method and provides enhanced accuracy relative to the classical proper orthogonal decomposition (POD) greedy method.