Discussions began last week in a bid to resolve the political row at Stormont.
The crisis was triggered when police said they believed IRA members had been involved in the murder of a former IRA man in Belfast last month.
Northern Ireland Secretary Theresa Villiers said finding resolution in the negotiations would be "very difficult".
But she added that the parties had "established a sound basis on which to take forward the discussions with greater intensity" this week.
She said the "impact of paramilitary organisations" was the matter on the agenda for Monday's round of talks.
That is one of the primary issues on the table during the cross-party negotiations.
Stormont's finances - a major factor is the implementation of welfare reform - is the other main matter on which the parties are trying to find resolution.
Ms Villiers had said the opening week of talks had been "useful and intensive" and that she was encouraged that the parties had shown a commitment to dealing with the impact and legacy of paramilitary activity.
In August, police said they believed IRA members had played a part in the murder of Kevin McGuigan Sr.
Chief Constable George Hamilton said the IRA was still in existence, but added that the organisation was not engaged in terrorism.
The claims led to a breakdown in trust between Sinn F??in and unionist parties.
The Ulster Unionists withdrew from Northern Ireland's ruling executive, with three Democratic Unionist Party (DUP) ministers also later resigning, and DUP leader Peter Robinson stepping aside as first minister.
Unionists agreed to join the talks when the government ordered an independent review of paramilitary groups in Northern Ireland earlier this month.

The African Elephant Status report says that poaching is the main driver of the fall, the worst losses in 25 years.
However the authors say that long-term issues such as the loss of habitat also pose a significant threat.
The report has been presented at the Cites meeting which is considering new proposals on elephant protection.
Extract from The War on Elephants; article on how the very existence of Africa's elephants is threatened by poachers
More from Alastair:
Why elephants are seeking refuge in Botswana
Shocking reduction in Africa's elephant numbers
Figures published earlier this year in the Great Elephant Census indicated that African elephant populations had declined by around 30% over the past seven years.
This new study from conservation group IUCN incorporates this information but also uses data from elephant dung counts and individual observations amongst other sources.
The authors say the overall total for elephants in Africa is now around 415,000, although there may be an additional 117,000 to 135,000 in areas not systematically surveyed.
This represents a decline of some 111,000 from the report carried out in 2006.
Poaching is the main driver of the drop. East Africa, the region most affected by killings for ivory, has experienced around a 50% reduction in numbers.
However it is not the only cause of concern to the authors.
"We are particularly concerned about major infrastructure projects that are cutting up the elephant ranges, this is a particular problem for road development in central and east Africa," said Dr Chris Thouless, one of the report's authors.
"These are all major issues that will have to be dealt with once the poaching crisis is over."
While report highlights the losses there are also some gains.
Elephant populations in South Africa, Namibia and Uganda have all increased.  Elephant ranges have expanded in Kenya and Botswana, with community based conservation showing real success in Northern Kenya.
While these are positives, the overall picture is one of dramatic decline, fuelled by criminal activity that would decimate these giant creatures, if continued.
"Larger quantities of illegal ivory are leaving Africa than ever before," said Ginette Hemley from WWF.
"The transnational crime syndicates driving the slaughter must be dismantled, and consumer demand for ivory cannot persist if we hope to secure a safe future for elephants."
The report comes as the Cites meeting here in Johannesburg is facing significant division on how to handle the poaching crisis.
Talks on extra protections for elephants will begin on Monday with a number of countries led by Kenya seeking extra protection.
Others, including Namibia and Zimbabwe, are seeking to liberalise the safeguards and open up a trade in ivory. Another proposal here, which might garner more support, is aimed at ending all domestic markets in ivory.  The meeting continues until 5 October.
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Talks between Northern Ireland's five main political parties have moved into a second week, with the focus on paramilitary groups.
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Elephant populations in Africa have declined by around 111,000 over the past 10 years according to a new study.