Many real-world optimization problems contain unknown parameters that must be predicted prior to solving. To train the predictive machine learning (ML) models involved, the commonly adopted approach focuses on maximizing predictive accuracy. However, this approach does not always lead to the minimization of the downstream task loss. Decision-focused learning (DFL) is a recently proposed paradigm whose goal is to train the ML model by directly minimizing the task loss. However, state-of-the-art DFL methods are limited by the assumptions they make about the structure of the optimization problem (e.g., that the problem is linear) and by the fact that can only predict parameters that appear in the objective function. In this work, we address these limitations by instead predicting \textit{distributions} over parameters and adopting score function gradient estimation (SFGE) to compute decision-focused updates to the predictive model, thereby widening the applicability of DFL. Our experiments show that by using SFGE we can: (1) deal with predictions that occur both in the objective function and in the constraints; and (2) effectively tackle two-stage stochastic optimization problems.