Parmar said getting children enrolled would ensure they were more likely to be immunised at six weeks, and any health or social issues would be picked up early.
At present, maternity carers send pre-enrolment forms to GPs on behalf of infants' families. Parmar said she understood that many GP clinics were acting too slowly in approving enrolments, and that was leading to delays in children getting immunised.
Parmar said her bill would put a firm timeline in place for doctors to follow.
If a practice was full, the GP would be required to help a family find another doctor. If no nearby GPs had room for the infant, the GP would have to refer the enrolment request to the relevant district health board to manage it.
The Government has a target of 95 per cent of eight month-olds getting their six-week, three-month and five-month immunisations on time. At last count, district health boards had achieved a rate of 93 per cent.
Labour's health spokesman David Clark said getting children enrolled early was a good idea, but the bill was toothless because it did require doctors or patients to comply with the deadline.
He said it appeared to add another administrative burden on GPs, who were already stretched, especially in Auckland.
At a health summit hosted by Labour at Parliament today, GPs raised concerns about resourcing, funding and the ageing workforce.
Responding to their concerns, Health Minister Jonathan Coleman said increasing GP numbers was a long-term challenge.
The Government was attempting to encourage medical students and young doctors into general practice by exposing them to rural GP roles at an early stage, he said.
The Ministry of Health was also phasing in a programme which required young doctors to spend time in the community as a requirement of their registration.
"So there is a long-term strategy and a focus on it, but of course it has always been challenging getting doctors to practice in remote rural areas," Coleman said.
Parmar's bill was one of four bills pulled from the ballot yesterday, and the only one likely to pass into law.
A bill in the name of Labour MP David Clark would prevent charter schools from making a profit, by requiring any income to be reinvested in education.
Labour MP Sue Moroney's bill would reverse a law change in 2014 which removed workers' rights to meal and rest breaks after regular periods of work.
Green MP Marama Davidson's bill would require councils to consider whether Maori wards should be created every six years - rather than deciding such an issue by referendum.
Davidson's bill was endorsed by former New Plymouth Mayor Andrew Judd, whose attempts to establish a Maori ward were defeated in a referendum.
