We present a numerical study of the order-parameter fluctuations for Ising spin glasses in three and four dimensions at very low temperatures and without an external field. Accurate measurements of two previously introduced parameters, A and G, show that the order parameter is not self-averaging, consistent with a zero-temperature thermal exponent value \theta' \simeq 0, and confirm the validity of the relation G=1/3 in the thermodynamic limit in the whole low-temperature phase, as predicted by stochastic stability arguments.