If your father lived past the late 1930's, he probably applied for a social security number at some point. His application should have his birth date (or at least the one he used) and his parents' name. A substitute for a birth certificate were religious records for infants (baptismal records or circumcision records), which often included in addition to at least one parent's name and the baptismal date, the date of birth.
The genealogy resource, FamilySearch, has a collection called " South Carolina, Delayed Birth Certificates, 1766-1900".
That collection can be searched for free, includes file images and contains birth records in the time period you need. Hopefully this link will work:
Dear Mr. Young,
Thank you for posting your request on History Hub!
To supplement what the previous posters suggested, the Social Security application for your father may be ordered here and the FamilySearch index for South Carolina, Delayed Birth Certificates, 1766-1900 is here.
We hope this is helpful. Best of luck with your family research!