Protesters remain on the Garvaghy Road in Portadown and Orangemen at Drumcree church as the parade dispute continues.
The main parties involved in the dispute in Northern Ireland are still in a stalemate, following a day of political talks and heated exchanges at Stormont.
Members of the Orange Order are still waiting to begin a parade, which was due to take place on Sunday, and walk down the mainly-nationalist Garvaghy Road.
The road was blocked by protesters and the parade did not go ahead.
Both the Orange Order and members of the Garvaghy Road Residents Coalition are meeting NI Secretary Sir Chris Bryant at Hillsborough Castle again on Tuesday.
Speaking on his way into talks with Sir Chris on Tuesday, Breandán Mac Cionnaith, from the Garvaghy Road Residents Coalition, said "reroute the sectarian parade, that's it".
"No parade is no parade, that's it," Mac Cionnaith said.
"That's what our community wants. That's what almost 4,000 people living in our community want and we gave that evidence to the Parades Commission last week in a signed petition."
Earlier, Orange Order Deputy District Master Alistair Power said he had told police they were ready to march down the Garvaghy Road whenever it was safe to do so.
Power said the "position of the district remains the same" and that the Parade Commission's determination is still "a live document", to bring the parade "down its traditional route of via the Garvaghy Road to complete its parade in Carleton Street".
About 30 to 50 protesters have been sitting roadside on a grass verge on the Garvaghy Road throughout most of Tuesday, but traffic has flowed freely.
A gazebo has been set up, with tea, coffee and snacks available and many of the residents and supporters have deck chairs.
A small group of about 10 people gathered at Drumcree church on Tuesday morning.
Taoiseach (Irish prime minister) Micheál Martin said he was in favour of an independent mediator to attempt to reach a solution.
"I understand fully that feelings are very high but this is a time for us all to be calm and to be careful in what we say and what we do," Martin told the Dáil (lower house of Irish parliament).
Sinn Féin leader Mary Lou McDonald and vice president Michelle O'Neill are meeting the taoiseach in Dublin.
McDonald said she will ask the Irish government to "intervene directly" in what she referred to a "sectarian march".
"You couldn't overstate the kind of trauma that has now been revisited on the community. Community life has been completely disrupted. People's family lives have been completely disrupted," McDonald said.
Christopher Nelson, who was 11 when his mother, the solicitor Rosemary Nelson, was murdered by loyalists in 1999, said the current situation had brought back memories of that time.
Rosemary Nelson represented Garvaghy Road Residents Association.
Nelson said he "dreaded the summer holidays approaching" because he knew his mother "would be spending a lot of time on the Garvaghy Road".
"With that came the worry about what was going to happen her when she was out there. Is mum coming home tonight and if she is, is she going to be okay?"
He said his mother spoke to her children about her job a lot and told them she was "helping families like our own".
"She tried her best to make you feel safe even though there was uncertainty in the back of your head," he told BBC Radio Ulster's Good Morning Ulster programme.
Nelson has never spoken publicly about his mother's death.
He feels now is the time to do so because "if she were here she would want me to be here and she would be standing on that road".
Now a father-of-two himself, and married to Sinn Féin councillor Catherine Nelson, he said he did not want any child "worrying or concerned" and said it was like it was "happening all over again".
He said his message to representatives was to "think about the children on both sides".
"This comes from a child who went through it, who experienced it, whose life it impacted. Do whatever needs to be done to protect them," he said.
He added that he was "very worried" about the current situation as "I know how badly it escalated and what the repercussions were in many cases".
The Drumcree parade, which in the 1990s sparked some of the most serious disorder in Northern Ireland's history, was due to go ahead for the first time in almost 30 years after a series of legal challenges.
Since 1998, the Orange Order had been banned from completing its Drumcree march - and until this year had been repeatedly refused permission to finish the route by the Parades Commission, which was set up to adjudicate disputes.
The Orange Order is a Protestant pro-British organisation, while the area where it wants to walk - specifically along the Garvaghy Road - is largely Catholic and predominantly inhabited by pro-Irish nationalists.
Protesters blocked the road over the weekend and, after meeting the Northern Ireland secretary, a representative of the road's residents said the Orange Order should focus on a different route.
On Sunday, protesters on the Garvaghy Road were joined by members of Sinn Féin, including First Minister Michelle O'Neill, and the Social Democratic Labour Party.
The Democratic Unionist Party subsequently accused O'Neill of acting unlawfully.
Sir Chris said "nobody should be attending an unlawful protest", but he added it was not his job "to decide whether somebody has or hasn't broken the law".
Speaking in the Northern Ireland Assembly on Monday, O'Neill said she was proud to protest with the residents.
Assistant Chief Constable Ryan Henderson said a "dedicated enquiry team" had been established, which was currently reviewing evidence gathered and "anyone who has broken the law will be investigated".
The Portadown District of the Orange Order has also lodged a complaint with the chief constable in relation to O'Neill and McDonald's attendance.
In a statement on Tuesday, the County Grand Orange Lodge of Belfast said it commended the Portadown District "for their dignity, patience and unwavering commitment to completing their lawful and traditional parade".
"The position of the County Grand Lodge is clear - there is a lawful Parades Commission determination in place. This has been upheld by the courts," it said.
"It is now for the PSNI to uphold the rule of law and facilitate the completion of the return home.
"Continued failure to implement a lawful determination risks even further undermining confidence in policing, which is already at an all-time low."
The Orange Order, the most prominent of the loyal orders associated with Protestants in Northern Ireland, holds thousands of parades every year and the vast majority pass off without incident.
However, many Catholics see the marches as sectarian and object to them passing through, or close to, Catholic or nationalist areas.
The Garvaghy Road had been part of the Drumcree route, but by the mid 1990s nationalist residents protested against the parade. The row sparked some of the most serious public disorder in Northern Ireland's history and was linked to several murders.
The row reemerged after a court case in which a ruling by the Parades Commission - the organisation formed during the Drumcree dispute to decide on contentious marches during Northern Ireland's parading season - was overturned.
On Friday, following 28 years of unsuccessful applications, the commission decided the march could go ahead with restrictions in place.
Rather than start at 13:00 on Sunday as intended, the commission said it could begin no later than 08:00 and with 35 participants rather than the 60 the Orange Order requested.
However that decision was challenged in several Belfast courts on Saturday, before finally being given the go ahead just hours before it was due to take place.
The Drumcree dispute was one of Northern Ireland's most serious parading disputes during the 1990s, when stand-offs over attempts by the Orange Order to march along the Garvaghy Road were accompanied by riots, roadblocks and killings.
When the Orange Order was prevented by police from walking down the route in July 1995, there was a tense stand-off. It was the same for the next three years.
Violence related to the dispute resulted in riots, roadblocks and murders across Northern Ireland.
In 1998, the newly-established Parades Commission ruled that the Orange Order should not march down the road - a decision upheld until Friday.
