In this paper, we propose the parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise. The proposed algorithms proceed as two-level temporal parallelizable integrators with the stochastic exponential integrator as the coarse $\mathscr{G}$-propagator and both the exact solution integrator and the stochastic exponential integrator as the fine $\mathscr{F}$-propagator. The mean-square convergence order of the proposed algorithms consistently increases to $k,$ regardless of whether the exact solution integrator or the stochastic exponential integrator is chosen as the fine $\mathscr{F}$-propagator. Several numerical experiments are illustrated in order to verify our theoretical findings for different choices of the iteration number k and the damping coefficient $σ.$