"The Waikato District Health Board have their own team of taxpayer-funded lawyers advising them and helping them to minimise the hospital's own liability," the petition says.
"Nicky's family want a level playing field. They want the DHB to fund their legal costs to the same level the taxpayer is funding the DHB's costs. Both parties should have have quality legal representation. Without equal legal assistance the family's voice may be drowned out by the DHB's parade of lawyers."
However, today the DHB declined the request, saying it would set a costly precedent.
The board made the decision after extensive consideration," chairman Bob Simcock said.
"If the board had agreed to pay the legal fees, it would set a precedent and result in cost implications for Waikato DHB," he said in a statement.
"We can't find any evidence of any other district health board making payments like this before. Meeting costs of this kind in similar cases would have an impact on patient care.
"Parliament's solution is to have a legal aid structure. If that is not a fair structure, the issue is for Parliament and not this board."
The board was following an internationally supported best practice quality process to look into the care provided by Waikato DHB to Mr Stevens, Mr Simcock said.
"To pay legal fees at this stage could have far-reaching implications for that process," he said.
The DHB is currently undertaking an internal review of the events leading up to Mr Stevens death.