If one of my divorced parents has been granted a green card but does not reside in the US (and has not resided there for longer than a few months), does that have any bearing on an application that I - a European citizen - make for residency?
It is unlikely that the fact that your parent holds a US green card will have any bearing on your application. In fact, if your parent does not currently reside in the US, their status of "permanent residency" could certainly be called into question (but also wouldn't affect your application).
This happened to somebody I knew in Canada many years ago. He was Canadian, was entering the US from Canada, and the immigration officer noticed his green card in his wallet. After some brief questioning, the officer established that he did not live in the US and cancelled his green card on the spot.