<p>In this paper, the optimal strong error estimates for the stochastic parabolic optimal control problem with additive noise and integral state constraint are derived based on implicit time discretization and finite element discretization. The continuous and discrete first-order optimality conditions are deduced by constructing the Lagrange functional, which contains forward-backward stochastic parabolic equations and a variational equation. The fully discrete version of forward-backward stochastic parabolic equations is introduced as an auxiliary problem and the optimal strong convergence orders in time and space are estimated, which further allows the optimal a priori error estimates for control, state, adjoint state and multiplier to be derived. Then, a simple and yet efficient gradient projection algorithm is proposed to solve stochastic control problem with integral state constraint and the convergence rate is proved. Numerical experiments are carried out to illustrate the theoretical findings.</p>