It's all covered under 14 CFR Part 21.
Production Certificates provide authorization for production to holders of a Type Certificate - typically the aircraft or engine manufacturer. Suppliers to the Production Certificate holder can produce components under one of two types of authorization.
If the part being manufactured is defined by the the OEM's Type Certificate (an OEM part numbered drawing), they can produce it under a PMA - Parts Manufacturing Approval. (14 CFR Part 21, Subpart K).
If the part is covered by a TSO - Technical Standard Order, the supplier can produce the part under a TSOA - TSO Authorization. (14 CFR Part 21, Subpart O).