City officials announced last week that an engineering problem will add as much as $600,000 in costs to the construction of the five-story deck.
The cost increases bring the total price of the deck to between $3 million and $3.1 million.
Earlier this year, the commissioners approved on a 3-2 vote giving $600,000 to the city to help pay for the project.
But after the cost hikes became known, city officials said they intend to ask the commissioners for another $300,000. The city will pay for $200,000 of the increased cost under an agreement with Bowman that was approved on a 4-1 vote this week.
While the formal request has not been sent to the commissioners, the two officials who opposed the city's original request remain opposed to the new request.
"If they're expecting us to pay an additional $300,000 - which would make a grand total of $900,000 - they're wrong," Commissioner John C. Munson said. "Six-hundred thousand's pretty fair."
Commissioners Vice President William J. Wivell said he also is opposed to funding the additional request. He said he believes the deck won't pay for itself and is an unaffordable luxury.
Parking garages are "something that you build if excess funds are available. I just don't see county government as having excess."
"We just can't afford everything. That's how we paint ourselves into corners and have to increase taxes," Wivell said.
Commissioners President Gregory I. Snook and Commissioner James F. Kercheval both said they believed the $300,000 request - which would not be given to the city for at least two years - was a worthwhile expenditure.
"We need to do this to help with the Arts and Entertainment District and ... the downtown parking issue. I think it's a good economic development project," Snook said.
Commissioner Doris J. Nipps on Thursday did not return phone calls seeking comment.