We do not currently know of any antonyms for third party.
The noun third party is defined as:
Someone not directly involved in a transaction. A third entity in the Seller (first party) and Customer (second party) relationship. A Seller may employ a third party to perform specific services to augment the value of a product. For example, a manufacturer may employ a third party to pack and distribute a product. A computer manufacturer may augment their product with software from a third-party supplier.