The decision follows a bitter dispute with Australia over which country should shoulder responsibility for the woman.
The woman had Australian and New Zealand citizenships, so Australia's action leaves Wellington responsible for her.
Family Frontiers criticises tedious application process and prolonged periods of limited access to education, healthcare and social protections.
They speak of bureaucratic dead ends and concerns for their children's future.
Lawyers for Liberty urges the government to amend the constitution to substitute the word 'father' for 'parent' in order to resolve the issue.