This article is devoted to the analysis of semilinear, parabolic, Stochastic Partial Differential Equations, with slow and fast time scales. Asymptotically, an averaging principle holds: the slow component converges to the solution of another semilinear, parabolic, SPDE, where the nonlinearity is averaged with respect to the invariant distribution of the fast process. We exhibit orders of convergence, in both strong and weak senses, in two relevant situations, depending on the spatial regularity of the fast process and on the covariance of the Wiener noise in the slow equation. In a very regular case, strong and weak orders are equal to $\frac12$ and $1$. In a less regular case, the weak order is also twice the strong order. This study extends previous results concerning weak rates of convergence, where either no stochastic forcing term was included in the slow equation, or the covariance of the noise was extremely regular. An efficient numerical scheme, based on Heterogeneous Multiscale Methods, is briefly discussed.