My son is 19 and was born in Germany while his father was stationed there with the US Military. He was issued a "Certification of Birth Abroad" by the Department of State. It was issued by the local Vice Consul in Germany.

The certificate has my sons name, his sex, place of birth, date of birth and the date the birth was recorded. It is stamped and signed by the Vice Consul at the time, John R Arndt. It does not mention his parents name at all.

My son needs to provide his birth certificate to his employer with proof of his parents name. Is there a legal document that we can provide? We are stumped on where to go for help.

No, why does he need "proof" of his parents names. But basically you can supply school records ( they are used for proof of ID)
But there is no "legal" reason anyone is required to prove who their parents are that I can think of.