We study the problem of post-processing a supervised machine-learned regressor to maximize fair binary classification at all decision thresholds. By decreasing the statistical distance between each group's score distributions, we show that we can increase fair performance across all thresholds at once, and that we can do so without a large decrease in accuracy. To this end, we introduce a formal measure of Distributional Parity, which captures the degree of similarity in the distributions of classifications for different protected groups. Our main result is to put forward a novel post-processing algorithm based on optimal transport, which provably maximizes Distributional Parity, thereby attaining common notions of group fairness like Equalized Odds or Equal Opportunity at all thresholds. We demonstrate on two fairness benchmarks that our technique works well empirically , while also outperforming and generalizing similar techniques from related work.