We consider a two-dimensional periodic reaction-diffusion system under natural conditions on the reaction function and with initial condition $θ$. We show that on the global attractor $\mathcal A$ of the resulting dynamical system $(u_θ(t):t>0)$, a reverse Poincaré inequality holds true, and that as a consequence the map $θ\mapsto u_θ(t)$ satisfies a $L^2$-Lipschitz stability estimate on $\mathcal A$ for any $t>0$ fixed. We then show that statistical recovery of an initial condition $θ$ in the attractor $\mathcal A$, as well as prediction of the states $u_θ$, is possible from discrete measurements of the system at `fast' near parametric convergence rates.