Her bill would lift leave to 26 weeks and allow them to work for up to 156 hours during that period without losing leave payments.
Last term, Finance Minister Bill English said the Government would use its financial veto to overrule any such measure if it passed because of the cost to the Government. On July 1 the Government increased paid parental leave from 14 to 16 weeks. A further lift to 18 weeks is due next year.
The other three bills drawn from the ballot were also Labour MPs' bills.
Labour social development spokeswoman Carmel Sepuloni's bill requires social workers to be registered and their qualifications reviewed every year.
Ms Sepuloni said there was currently no such requirement for social workers, which meant anyone could claim to be a social worker.
"This is dangerous and puts children at risk."
Two other bills target the charter schools, including Phil Goff's bill to require charter schools to use the same curriculum as state schools.
Adrian Rurawhe's bill targets Act leader David Seymour, who as an Under-secretary for Education, has some responsibilities for the 'partnership schools' which National agreed to introduce under its support agreement with Act.
Mr Rurawhe's bill would make under-secretaries subject to the Official Information Act. Currently only ministers are subject to the act -- normal MPs and Parliamentary Service are not.