There is indeed implication that the Gerudo were involved. In Hyrule Historia it is mentioned Ganondorf 'once again' swears fealty to the King of Hyrule. Which implies that he was loyal before the war and fought against the king during.
Since Ganondorf is King of The Gerudo if he was not on the...
I tend to think that the King of Hyrule just despatately didn't want to have another war and had convinced himself that confronting Ganondorf would ruin all his efforts or convinced himself that Ganondorf wanted peace too and therefore didn't want to hear anyone suggest otherwise.
He was the...