The parties have been holding talks at Stormont Castle
Talks about the devolution of policing and justice powers to the Northern Ireland administration at Stormont Castle have ended.
DUP and Sinn Fein negotiating teams met for most of Wednesday in an attempt to resolve the impasse.
Martin McGuinness will give a report on the state of the negotiations to the Sinn Fein officer board on Thursday.
BBC political reporter Stephen Walker said parading was one of the big issues discussed.
It was on a position paper the parties had been studying.
"A source told me they're going through each issue one by one," he said.
On Wednesday, SDLP MLA Alex Attwood said the personal difficulties surrounding DUP and SF leaders Peter Robinson and Gerry Adams must not get in the way of finding a solution.
He said "whatever the fog may be" it was now essential that the devolution of justice was "up and running".
Unlike talks on Tuesday, the other Northern Ireland political parties have not been involved.
Earlier, it emerged that senior UUP and DUP figures held private talks at the weekend.
Attended by Mr Robinson, the talks were hosted in England by Conservatives who said they aimed to "promote greater political stability".
Party sources denied they were about an electoral pact or voting arrangements in the event of a hung parliament.
UUP deputy leader Danny Kennedy said they were "not secret talks but political discussions" which would be "welcomed by pro-union people".
"We do not propose to go into any detail, except to say that we considered the short-term, medium-term and long-term interests of all the people of Northern Ireland," he said.
The parties have been at loggerheads over policing powers
Meanwhile, the Ulster Unionists' only MP, Lady Sylvia Hermon, has said she wants clarification over a special meeting of the party executive.
The meeting at the weekend will hear an update from party leader Sir Reg Empey on recent political negotiations and possibly take decisions about joint UUP-Conservative candidates for the upcoming general election.
However, Lady Hermon, who opposes the UUP's electoral pact with the Conservatives, says she may attend as she wants to hear the party's plans for her North Down constituency - the only one for which the party has so far failed to call a candidate selection hearing.
SDLP leader Mark Durkan warned the Tories against "distracting" the unionists from the current talks on the devolution of policing and justice powers.
"I think, at a time when everybody should be concentrating on the big issue that is in front of us, it's unhelpful for the opposition party in Westminster, who are presenting themselves as the government-in-waiting, to distract parties in this way," he said.
The Conservative Party denied suggestions that it discussed the consequences of a hung parliament with the two main unionist parties and the discussions were focused on the current difficulties at Stormont.
The spokesman said: "So far as the Westminster election is concerned, the only deal is the current deal between the Conservatives and the Ulster Unionists.
"We will be putting up 18 Conservative and Unionist candidates at the next election," he said.