He told the Herald over the weekend despite the introduction of a voluntary code of practice in 2015, the issue of clamping was "the same as it's always been".
"I think too often there's a temptation to use [clamping] as a money-making exercise," he said.
Hefty fees were "totally out of proportion" for a few minutes of parking, even if the driver was in the wrong.
Goff would be seeking responses from council to see if the bill has wider support once Wood had cleared its submission with the clerk's office, he said.
Labour has also submitted a private member's bill seeking to make the current voluntary code of conduct which regulates parking enforcement compulsory - but it's been in the ballot for more than a year.
Local bills avoids the usual members' ballot and have priority in Parliament over ordinary members' bills.
Goff hoped the act of submitting a local bill would be an incentive for consumer affairs Minister Jacqui Dean to act faster on a nationwide solution for the issue.
Dean told the Herald she was waiting for advice from officials about wheel clamping but had not had time to look into it yet.
National MP for Auckland central Nikki Kaye has also said she will be looking into clamping.
Wood said he believed the minister was choosing not to prioritise the issue, despite the "abuse and intimidation" drivers were suffering at the hands of some clampers.
"I would have thought it would be enough of a kick that the minister would take a bit of sharp action."
The public deserved quick and energetic action around consumer issues, Wood said, and he was concerned the minister was paying lip service to change without intending to follow through.
"One of the concerns with these sorts of issues is that ministers say 'yup we're looking at it' and that's a holding pattern until interest drops off, and nothing happens."
While the minister was new to the role, having taken on the portfolio last December, Wood said so far he wasn't impressed with her handling of consumer issues.
"There have been two serious issues that have come up in that time - free-range eggs, and this issue, and there isn't exactly a lightning speed action to either of them."
Dean did not respond to request for comment in time for deadline.