The optimal solution for caching POST requests is to use API Gateway’s built-in caching capabilities. API Gateway allows for the caching of responses to POST requests by configuring caching at the stage level. This leverages the native functionality of API Gateway to manage caching efficiently without requiring significant changes to the application logic. Options A, C, and D are not suitable because CloudFront does not cache POST requests, and using Lambda's /tmp directory or AWS Systems Manager Parameter Store for caching responses is not as efficient or direct a solution as using API Gateway’s caching features.