The Orange Order plans to only play hymns during its Ulster Covenant parade past a Catholic church in the Donegall Street later this month.
However, they have refused to meet a local residents' group to discuss the march, arguing that locals could have joined in with their discussions with the Catholic Church.
The motion was brought to the floor of the house at Stormont on Tuesday - and rejected.
A UUP amendment to the motion, calling for recognition of the "positive community contribution by the Loyal Orders" was passed by 50 votes to 48.
UUP MLA Tom Elliott has called on republican and nationalist politicians "to accept the progress that has been made instead of continuing to inflame the situation."
He added: "It is imperative on all of us to ensure that we promote a less divisive approach to parades and other events that may be deemed contentious.
"We also need to accept that not everyone will have a similar viewpoint on these matters, how they are progressed or what the outworkings of such will be."
The Parades Commission met with concerned parties and residents on Tuesday to discuss the upcoming parade, which will pass St Patrick's Catholic Church.
Afterwards, Frank Dempsey from Carrick Hill Concerned Residents Committee said that residents had no objections to parades, but asked for respect their community and church.
"This empty gesture by the Orange Order in no way addresses any of these two issues," he said.
"It's simply a ploy by the Orange Order to bypass the Parades Commission and the Carrick Hill Concerned Residents Committee, in order that the previous determination is overturned to facilitate the Ulster Covenant parade."
Sinn Féin MLA Gerry Kelly said that there should be only a drumbeat or no music played by bands during that part of the parade.
Father Michael Sheehan from St Patrick's said he had been encouraged by initial talks, but ultimately expressed disappointment that the Orange Order had not included Carrickhill residents in talks.
"I am bewildered that they are being excluded from 'the conversations'," he said.
The priest described the decision to only play hymns as an improvement to the actions that occurred during the Royal Black Preceptory event, but added that it was "no substitute for real and meaningful dialogue".
SDLP MLA Alban Maginness reiterated a call for the Orange Order to engage in direct talks with north Belfast residents ahead of the parade later this month.
He told UTV he hoped the order would make an effort to have dialogue with the "whole community", outside of their talks with parishioners at St Patrick's Church.
"The Orange Order has been making moves in the right direction when it comes to easing tensions in north Belfast, but talking with residents is a critical and essential element if there is to be a genuine, mutually respectful process," he said.
At the meeting with the Parades Commission, Mr Maginness said he believed that no music at all should be played outside the church.
But the decision to play hymns has been welcomed by unionist politicians as a gesture of "goodwill" from the Grand Orange Lodge.
UUP leader Mike Nesbitt said he hoped the "reasonable" move would mean the event would take place "peacefully and respectfully".
It is the Orange Order and the bands who have taken the first step - somebody else needs to come in and reciprocate.
Mike Nesbitt, UUP
He added: "Is there anything wrong with a band playing a hymn on a street in Belfast on a Saturday?
"The answer for me is no. And if that's the case, then the question for the Parades Commission is: 'Is it reasonable to place any further restrictions on the parade?'
"And again, I think the answer would be no."
DUP MP Nigel Dodds added:"The focus should return to the purpose of the parade, which is the celebration of an enormously significant event in the history of Ulster."
In August, a Parades Commission ruling that bands should only play a single drumbeat when passing the church was defied by bands during a Royal Black Preceptory event.
That followed an earlier incident, in which a loyalist band was filmed playing an alleged sectarian song outside the same church on the Twelfth of July.