Portadown Orange Lodge at police lines in 2009
The Garvaghy Road dispute can be resolved by the Orange Order never marching there again, the Deputy First Minister has said.
Martin McGuinness said he believed it was "not too much to ask" to expect that the Orange Order acquiesce to resident demands not to walk there.
He was speaking after DUP MLA Nelson McCausland accused Garvaghy Road residents of "cultural apartheid".
He said they were guilty of attempting to take ownership of certain roads.
Orangemen have been banned from walking down the road during their annual march from Drumcree church since 1998.
Following the Hillsborough Agreement, the DUP and Sinn Fein have set up a six-strong group - three each from their respective parties - to try to agree a framework for dealing with parades.
The group, which is completing its first week of discussions, is due to meet with the SDLP on Friday.
Mr McGuinness said that the unionist community and Orange Order should "make a gesture" to residents of the road by declining to walk down it.
He added: "I think it is not too much to ask given the trouble we have seen on the Garvaghy Road."
But Mr McCausland said residents "were trying to claim ownership of particular roads and say "you cannot go there without our consent".
"There is no other way to describe that but cultural apartheid," he said.
Speaking before the meeting the parades working group, SDLP MLA Dolores Kelly said that her party had been "steadfast" in wanting to see the Garvaghy dispute resolved through "local agreement".
She said the SDLP delegation would be telling the group that its framework should mean the retention of the Parades Commission.
"We want to see the Parades Commission plus, not the Parades Commission minus," she added.
Brendan MacCionnaith, spokesperson for the Garvaghy Road Residents Coalition, said that Orangemen could take the same route back from the church as the one they take to get there.
He said that if there was going to be dialogue "then all the options had to be on the table".
"The right to freedom of assembly does not equate to an absolute right to march," he said.
Mr McCausland, Northern Ireland's culture minister, was speaking after a DUP delegation met the Parades Commission, which rules on contentious parades, on Thursday.
In a statement the commission described the meeting as "a positive and useful exchange".
"We urge everyone in a position of influence to promote and encourage dialogue in Portadown and wherever parades are disputed," it said in a statement.
"The commission hopes that all parties in Drumcree will see the merit of dialogue and agree that it is the best way forward."
The last time Orangemen paraded along the Garvaghy Road was in 1997.