We consider the problem of regression under the demographic parity constraint without access to the sensitive attribute at the inference time. We build a general-purpose post-processing algorithm that, given an accurate estimates of the regression function and sensitive attribute predictor, outputs a prediction function satisfying the demographic parity constraint. The algorithm relies on discretization and on a stochastic minimization of a convex smooth function. It can be used for online post-processing and for addressing multi-class classification setup. Unlike previous algorithms, we provide a fully theory-driven approach. We require a tight control of the norm of the gradient of the aforementioned convex function and thus rely on more sophisticated methods than the vanilla stochastic gradient descent. Our algorithm is supported by a finite-sample analysis and post-processing type bounds. Experimental validation confirms the developed theory.