Twelve women working as staff in a North Korean restaurant in China arrived in the South, along with their manager, earlier this month.
Seoul said they came voluntarily, while the North insists they were tricked into defecting by South Korean spies who effectively "kidnapped" them with the connivance of the manager.
A spokesman for the North Korean Red Cross said the parents of the 12 staff were demanding "direct contact" with them as early as possible.
Panmunjom is the UN truce village situated on the inter-Korean border.
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"What we want is to let the daughters meet their parents and directly clarify their stand," the statement said.
A refusal by Seoul would be tantamount to "self-admitting the group abduction," it added.
Nearly 30,000 North Koreans have fled poverty and repression at home to settle in the capitalist South.
But group defections are rare, especially by staff who work in the North Korea-themed restaurants overseas that are a key source of hard currency for the regime in Pyongyang.
Anyone caught fleeing the country can be subject to harsh punishment, as can the families of those who successfully defect.
Relatives are often featured in state propaganda, either making tearful pleas for defectors to return home or berating them for betraying the motherland.
The latest defections came at a time of elevated military tensions on the divided Korean peninsula following Pyongyang's nuclear test in January.
North Korea is scheduled to hold a rare party congress in May - aimed at showcasing the country's achievements and fostering a sense of national loyalty and pride.