Does Portuguese language distinguish between the word for anything related to a country/nation and people of that country/nation, eg. Spanish vs. Spaniard?
We use the same word very often. If there is a different one, it's pretty rare.
Could you say parents instead of pais
No, parente(s) in Portuguese is any relative, not just parent(s).
In the previous exercise it was minhas primas sao espsnholas but espanhois was "wrong" but in this sentence they use it, the only difference is cousins/parents???
One is feminine the other isn't. "Minhas primas" refers to female cousins so naturally the adjective (espanhol) would be feminine too (espanhola)
Pais on the other hand is a masculine noun so the adjective natually follows its gender.