Exactly what is a "third party"? (And who are the first and second party?)
I know precisely what a "third-party library" is, so I assume that the "third-party" in this case, simply is an arbitrary person/company other than the developer?
Has it to do with "first person" / "second person" etc perhaps?
The fact that there is a "third party" suggests that there is a "first party" and a "second party" as well. Are those terms well defined?
(I'm not a native english speaker.)
Answer
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.