The town hall had hoped its own planning committee would give the green light to a bid for land between Penny Street and Hardwicke Street.
But the proposal has been deferred after councillors heard concerns from neighbouring residents.
The delay will come as a blow to the local authority as the new facility has been designed to mitigate the loss of the market multi-storey car park, part of the multi-million pound Markets Quarter redevelopment.
Officers said it would bring a disused site back into use.
However, committee members heard how noise and pollution would impact on the quality of life for residents of St Ignatius Square, whose properties back onto the land.
The council’s plan includes a shin-high fence running along the perimeter that residents said would be inadequate.
Ward councillor James Hull called on the committee to defer the decision. He said the residents’ group was not against the car park in principle but urged that design changes should be made.
The residents’ group called for a reduction from 76 to 59 spaces, an increased distance between the rear of the properties and the parking bays in addition to a chain-link fence and landscaping features.
Friends of St Ignatius Square representative Nick Millband told the committee: “We’re not asking for much. It’s a conservation zone, three years is a long time for a conservation area not to have any landscaping, it’s a totally inappropriate usage. Put some trees in. Are 17 car parking spaces worth more than the health and wellbeing of residents?”