Solute strengthening is an important mechanism that contributes to improving the mechanical properties of alloys and particularly the recent generations of concentrated alloys. The stress field emerging from an elastic model of a random solid solution displays strongly anisotropic correlations that interact differently with dislocations of different characters. In the present work, we investigate the depinning transition of edge and screw dislocations evolving in such a correlated stress environment using a dislocation dynamics numerical model. We find that edge dislocations are only weakly affected by the correlations, while screw dislocations are strongly influenced, showing a smaller critical stress, which increases with the amplitude of the stress noise with a larger exponent than the edge dislocation. The numerical results are compared with existing statistical models of solute strengthening, allowing to discuss critically their assumptions.