Is it permissible for a wife to undergo medical tests without her husband's knowledge if he insists on informing his parents of all details, and is it her right to have privacy?

Your husband's statement that he must inform his parents of all his private affairs is incorrect, and you have the right not to let anyone know your private matters. There is no objection to undergoing medical tests without your husband's knowledge if he insists on informing his family about them and you are harmed by that, provided that it does not involve a Shariah violation or corruption.