North Korean leader Kim Jong-un had asked Moon to tell the pope of the invitation.
The visit was suggested by Mr Kim last month at an inter-Korean summit aimed at finding a way to reach lasting peace on the peninsula. The Pope's office reported that he had said: "If the invitation comes I will surely respond to it, and I can possibly go".
Kim told Moon, a Catholic, of his wish to meet the pontiff during a meeting last month and the South Korean leader announced before the trip that he would be relaying a message.
A trip that would be the first by a pope to officially atheist North Korea would need "serious preparation and consideration", Mr Parolin said.
South Korean President Moon Jae-in delivers a speech at the end of a Mass for peace inside St. Peter's Basilica at the Vatican, Wednesday, Oct. 17, 2018.
Earlier Thursday, Moon wrapped up his three-day visit to Italy and the Vatican.
The Vatican's priests were expelled by North Korea long ago and state-appointed laymen officiate services. Moon's office quoted the pontiff telling Moon: "Do not stop, move forward".
It spoke only of "the promotion of dialogue and reconciliation between Koreans" and 'the common commitment to fostering all useful initiatives to overcome the tensions that still exist in the Korean Peninsula, in order to usher in a new season of peace and development'.
US officials are concerned that the no-fly zone proposal, which is part of an agreement between North and South Korea reached during a summit last month in Pyongyang, could hinder defense readiness, according to Reuters, which cited two unidentified sources. Previous popes had refused to cut a deal with China's communist leaders, who allow religious practice only in state-sanctioned churches.
Following an unusually provocative run of weapons tests a year ago, Kim has been on a diplomatic offensive, which included three summits with Moon and the one with Trump.
"The two presidents discussed a wide range of issues, including the countries' diplomatic and defense cooperation, ways boost their future-oriented cooperation, conditions on the Korean Peninsula and their global cooperation", Moon's presidential office Cheong Wa Dae said.
Aides close to the pope have said he is open to taking what they call first steps in places where the Church has been persecuted in the hope that the situation could improve.