We study the compatibility between the optimal statistical parity solutions and individual fairness. While individual fairness seeks to treat similar individuals similarly, optimal statistical parity aims to provide similar treatment to individuals who share relative similarity within their respective sensitive groups. The two fairness perspectives, while both desirable from a fairness perspective, often come into conflict in applications. Our goal in this work is to analyze the existence of this conflict and its potential solution. In particular, we establish sufficient (sharp) conditions for the compatibility between the optimal (post-processing) statistical parity $L^2$ learning and the ($K$-Lipschitz or $(ε,δ)$) individual fairness requirements. Furthermore, when there exists a conflict between the two, we first relax the former to the Pareto frontier (or equivalently the optimal trade-off) between $L^2$ error and statistical disparity, and then analyze the compatibility between the frontier and the individual fairness requirements. Our analysis identifies regions along the Pareto frontier that satisfy individual fairness requirements. (Lastly, we provide individual fairness guarantees for the composition of a trained model and the optimal post-processing step so that one can determine the compatibility of the post-processed model.) This provides practitioners with a valuable approach to attain Pareto optimality for statistical parity while adhering to the constraints of individual fairness.