If you are developing software for a client, then there is a contract between you/your company, and the client/their company. These are the two parties to the contract. Anyone else, not bound by the contract, is a third party. It’s used wherever a contract exists between two parties to mean anyone not bound by the contract.
There is no fixed meaning to which of the two parties is ‘first’ and which ‘second’, usually you will think you’re the first party, and the client the second, whereas the client will think they are the first party and you the second, in a similar fashion to first, second and third person I/he/they.