The Ministry of Gender, Children and Social Protection has also been hauled to appear before Parliament.
The petition forms part of a campaign by Child Rights International (CRI) to mount pressure on the appropriate agencies to bring back the missing girls.
The Executive Director of CRI, Bright Appiah said the organisation would use every available avenue to ensure that the case involving the three missing girls was given its deserved attention.
"The organisation is presently trying to secure more signatories from Ghanaians and send it to Parliament to demand accountability on the missing girls," he said in a statement.
He also urged the government to look beyond the law and consider the social cost on the families.
"Unfortunately the focus on the part of the state has been on the criminal side to the neglect of its social obligations to the family," the statement added.
Ruth Quayson, Priscilla Blessing Bentum and Priscilla Koranchie were reportedly kidnapped between August 2018 and January 2019.
The Police has so far managed to arrest the suspect, identified as Nigerian national Samuel Wilson Udoterg, but have not been able to extract much from him.
In April, the Police CID boss at a press conference disclosed that they had discovered the whereabouts of the girls.
COP Maame Yaa Tiwaa Addo-Danquah said the police was putting measures in place to hand them to their families.
However, nothing has since been heard after that press conference, with parents of the kidnapped girls still soliciting answers from the police.