A number of senators said the meeting was in contrast to the interests of the Afghan people.
Some MPs said the rift should end as soon as possible because it has affected the activities of the parliament.
The Acting Minister of Interior says the case has many dimensions and needs a thorough investigation.
A former Taliban member says timetable for troop withdrawal is key to progress in US-Taliban talks.
The provincial police spokesman says the incident took place in the city of Taluqan on Saturday evening.
Officials said the incident took place in Khoshal Khan area in Kabul on Sunday morning.
Khalilzad says that he believes all sides want "rapid progress" in the peace process.
Head of the peace marchers said they were not allowed by the Taliban to meet with people living in their areas.
Qureshi says they want to see a friendly Afghanistan, governed by an elected leadership.
Pakistani Foreign Minister Shah Mahmood Qureshi spoke at the opening of the conference.
Sign up to receive the best of Tolo News daily