Is what the questioner did—registering his mother in her name to obtain his father’s medicine from the company’s insurance, before his father was included in the insurance after reaching the age of sixty—considered unlawful, and what is required of him to atone for that?

There is no harm in you obtaining your father's right to treatment at the company's expense through the mentioned stratagem, as long as you were unable to officially prove it, and as long as you fulfilled the agreed-upon condition that your father not be covered by external health insurance. The stratagem is permissible if it is necessary to obtain a right and does not result in harm to others.