The negotiations in Moscow follow recent talks between the U.S. and the Taliban in Qatar.
Talks have mostly focused on a US troop withdrawal and guarantees from the Taliban that Afghanistan would not again be used as a safe haven for terrorists to attack other countries, according to both Khalilzad and Taliban officials.
The meeting has sidelined Afghan President Ashraf Ghani's government, which has criticized the gathering.
That may now change after U.S. President Donald Trump said he wanted to eventually pull American troops out of the country and end almost two decades of U.S. military involvement there.
On the critical issue of women's rights, Stanikzai said the Taliban were committed to all rights of women "that have been given to them by the sacred religion of Islam".
Taliban deputy chief negotiator, Salam Hanafi, in Moscow once again publicly says Americans agreed in Doha to withdraw half their troops by end of April. The government in Kabul said the talks were not in Afghanistan's best interest.
"We do not know whether we will achieve an agreement - but we do know that after two decades of war, the hour has come to at least try for peace", Trump said. While the Afghan population is exhausted and traumatized, and the Trump administration is seeking to bring the conflict to a close as part of a campaign promise to reduce USA troops overseas, the insurgents control territory that over 10 percent of the population inhabits.
US-led forces in 2001 toppled the hardline Taliban for harboring the al Qaeda militants responsible for the September 11 attacks.
Ghani's call to Pompeo seemed an attempt to re-assert his government's role in negotiations he has sharply criticised as one-sided.
The Kremlin hosted another Taliban meeting in November previous year which brought to the negotiating table representatives of the insurgent group as well as members of the Afghan peace council - a body created by the Afghan government to facilitate peace talks.
Two days of unprecedented talks in Moscow with high-ranking Afghan politicians end with disagreements over women's rights and demands for a new constitution..
Taliban representatives and an Afghan delegation led by former President Hamid Karzai have held a second and final day of talks in Moscow, with the militants continuing to insist upon the withdrawal of all foreign troops from Afghanistan as a first step in the peace process.
Mr Stanikzai told the BBC he believed the Trump administration wanted to "bring peace to Afghanistan".
Karzai led a delegation of about 40 Afghan politicians - although without representatives of the current government in Kabul - at the two-day round of talks in Moscow on February 5-6.
"They can go to school, they can go to universities, they can work", he added. In a lengthy interview with the privately owned ToloNews TV, Ghani denounced the ongoing Moscow meeting, arguing the attendees did not have the executive power to translate decisions into action.
"We agreed to continue talks between the Taliban and the United States", Atmar said. "Let hundreds of such meetings be held, but these would only be paper (agreements) unless there is an agreement by the Afghan government; Afghanistan's national assembly and Afghanistan's legal institutions".
"We are happy with the outcome of the meeting", Karzai said.
The Taliban have been staging near-daily attacks, inflicting heavy casualties on the embattled Afghan army and security forces.