Potential asbestos exposure from adding screws to drywall?
IMHO Asbestos dangers have been dramatically over-stated for the general public. The people who were in real danger are  people who worked with asbestos in an industrial setting, like ship workers insulating pipes, heating techs insulating pipes and ducts with asbestos etc. , day in and day out, cutting it, being exposed to it's dust.
Extremely limited exposure to asbestos like your father may have had will not result in any health problems.
Hopefully this answer will not get flagged for being opinion based because if you look at the facts, nearly everyone who got asbestosis was in an industrial or military setting.   Your father is at virtually no risk from what he did.