We address the regression problem under the constraint of demographic parity, a commonly used fairness definition. Recent studies have revealed fair minimax optimal regression algorithms, the most accurate algorithms that adhere to the fairness constraint. However, these analyses are tightly coupled with specific data generation models. In this paper, we provide meta-theorems that can be applied to various situations to validate the fair minimax optimality of the corresponding regression algorithms. Furthermore, we demonstrate that fair minimax optimal regression can be achieved through post-processing methods, allowing researchers and practitioners to focus on improving conventional regression techniques, which can then be efficiently adapted for fair regression.