He spoke out after meeting residents of the nationalist Garvaghy Road in Portadown, Co Armagh, where Orangemen marched last summer after a three- day stand-off with police at Drumcree.
This year, Mr Adams said: "Nationalists are not prepared to let the Orange Order walk over them. The days of Orange supremacy are long gone." He said the local MP, Ulster Unionist leader David Trimble, could show leadership by engaging with his own constituents.
Mr Adams added: "The onus however, should not be on the people of the area but on the two governments to uphold their rights and to prevent a repetition of the disgraceful events of last year."
But the Irish Prime Minister John Bruton, speaking on a visit to Northern Ireland, said problems over parades were best resolved at local level.
He declined to comment on a call by Mr Adams earlier this week for Irish government monitors to be sent to all flashpoint parades.
Meanwhile, a major security operation was mounted in Londonderry last night when police and troops launched a hunt for an IRA bomb. The "explosive device" may have been abandoned in the nationalist Creggan area up to three days ago, said the RUC.