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.)
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 meaning given to 'first' and 'second' party.
(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)
It's a term that's often used in Windows-centric development: the first and second parties are me (or you), and Microsoft; and the third party is anyone else:
Sometimes it means the customer or end user (e.g. "if we get a 'redistributable' from Microsoft, that means that we can redistribute it to 'third parties'")
More often, it means a non-Microsoft vendor of programming tools or libraries, which I'm using (for example, "NUnit and Reflector are both example of 'third-party' tools").
I think of it as from where the code comes from, so when it comes to libraries and development tools I'd say that the first and second parties are the developer and the producer of the development tool. So as a .Net developer the parties are me and Microsoft, since I write code using Microsofts framework and controls and I might also then user third party code/controls.
These terms are well defined in English when talking about grammar (English grammar or another language's).
First person corresponds to the pronouns "I" and "we"; "me" and "us" (so a book written in the first person is a story told by the central character - "I did this" as opposed to "Smith did this").
Second person corresponds to the pronoun "you".
Third person corresponds to the pronouns" he", "she", "it" and "they"; "him", "her" and "them".
So "third party" just means not you or me, but them.
I don't think first and second party are used that much, if at all, in programming. If someone started talking about first and second parties at work, I would wonder what they meant. However, "third party" is much more common and familiar.
"Third party software" is a common term. I've never heard of "first party software", or "second party software".
PS: I'm a native English speaker in the UK.