Parameter estimation for a parabolic linear stochastic partial differential equation in one space dimension is studied observing the solution field on a discrete grid in a fixed bounded domain. Considering an infill asymptotic regime in both coordinates, we prove central limit theorems for realized quadratic variations based on temporal and spatial increments as well as on double increments in time and space... Resulting method of moments estimators for the diffusivity and the volatility parameter inherit the asymptotic normality and can be constructed robustly with respect to the sampling frequencies in time and space. Upper and lower bounds reveal that in general the optimal convergence rate for joint estimation of the parameters is slower than the usual parametric rate. The theoretical results are illustrated in a numerical example. (read more)