The news came as the SNP-led Scottish Government announced new laws permitting councils to introduce workplace parking levies would come into effect in March.
Scottish Labour said its election manifesto for the city council had not been published yet, but “we understand this won’t be in it”.
A spokesperson said: “The case against it has changed over time – cuts to council budgets increase the risk that it will be used inappropriately, and pressures on households budgets have gotten worse.”
The levy is included in the council’s City Mobility Plan, which stated a consultation into the scheme would be completed by next year.
It would see employers paying an annual levy on every parking space at their premises, which they could pass onto staff and visitors.
The money raised would be ring fenced for improving public transport, cycling and walking.
The scheme would not cover shopping centres or retail parks, but councils could choose to include schools, colleges and universities.
Other exemptions include blue badge holders’ spaces, those for healthcare workers at hospitals, and parking at hospices.
The only scheme to be introduced in the UK is in Nottingham, where employers with 11 or more spaces are charged £428 a year per space.
Scottish Labour transport spokesperson Neil Bibby criticised the decision to press ahead with the plans.
He said: “This SNP commuter tax is a shameless attack on workers’ pay packets.
“This will hit countless front line workers who kept us going through the pandemic.
"It beggars belief that the SNP are pressing ahead with these misguided plans in the midst of a cost of living crisis.
“We need to reduce car use by building an affordable, reliable and accessible public transport – not by forcing people to pay for the privilege of going to work."
Edinburgh council’s SNP transport convener Lesley Macinnes said: “Now guidance has been published, we will be considering this carefully before bringing forward any proposals, which would be subject to widespread public engagement before any changes are made.
“We need to be open to a number of radical measures which will help us to tackle the huge contribution that transport makes to health-related issues as well as air pollution and city congestion.”
SNP-led Glasgow City Council said it was also planning to introduce a scheme, but no proposals had been published.
City convener for sustainability and carbon reduction Anna Richardson said: “A licensing scheme for workplace parking could help to raise substantial funds for the sustainable transport projects needed to meaningfully address climate change.
“It would incentivise employers to encourage greater use of sustainable transport as well as reduce car journeys.
“In a city where almost half of our households do not have access to a private vehicle, a workplace scheme would contribute to a more equitable and efficient transport system.
“The council is currently exploring the potential for a scheme and we will be looking at the new regulations very closely before bringing forward any proposals.”