The petition, or Early Day Motion, was submitted to the House of Commons last week and the union is urging UK members and their families and friends to write to their MP asking for backing.
In a release from the union, NUJ freelance organiser John Toner said he was delighted the idea was being discussed in parliament.
"We need to secure enough signatures to make the government take notice," he said.
At the time of writing, 14 MPs had signed the EDM, which asks for freelance workers to be made eligible for redundancy pay in the same way that they qualify for pro rata holiday pay.
If approved, the improved rights would benefit not only journalists, but workers across all sectors where regular freelance work occurs, the union has said.
A report from the All-Party Parliamentary Group for the Freelance Sector, including evidence from the NUJ and broadcasting union BECTU, has also called for more protection for freelance workers, the NUJ said.
The group's report said: "We are concerned at the plight of those in the media industry who are called freelance, or 'casual', when in fact they would wish for permanent positions.
"We would like to see the definition of the word 'worker' broadened to include all dependent workers and that these workers should have a statutory entitlement to workers' rights on a pro rate basis."
The NUJ has declared July 'Freelance Month' to highlight the plight of freelances, who, according to NUJ general secretary Jeremy Dear, 'have been hit by the media cut backs as hard as staff [workers]'.
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