The council has yet to announce when the feedback period opens.
Parking costs will still increase to $1.70 an hour, an increase of 20 cents.
"This will require us to re-programme the parking meters," chief financial officer Stuart McKinnon says. "We will announce shortly what date the fee increase will come into effect."
At the meeting, Smith said with the level of feeling that came through about the increased hours, not only in the petition but also emails and calls, he thought the council had got the change "slightly wrong", including the process.
"There's no harm in admitting that it has been done wrong."
It was evident the council had passed something that had upset thousands of its residents, Smith said.
"This is about the people rather than the process. It's caught me by surprise and I admit that as mayor and I know it has caught some of you as councillors by surprise."
The council agreed to ask the chief executive to undertake "stakeholder engagement" on the on-street parking time changes and report back to the September council meeting.
Councillor Susan Baty, who chaired the March Finance and Audit Committee meeting at which the parking changes were recommended, said she was "very, very surprised" there was no questions asked at that meeting.
Baty said members of the public had approached her saying staff had done things councillors did not know about.
"We as councillors passed this, the staff make the recommendations from a workshop, so I do not want staff hung out to dry for this."