One woman, who did not want to be named, said it was always hard to find parking spaces around the park and parents had to park where they could.
"It feels a bit like entrapment. We have to bring the kids here to play, there are not enough places to park for all the parents and then they hit us with tickets. It's like shooting fish in a barrel for them," she said.
But the council is staying tough, saying there are plenty of parking spaces just a few minutes' walk away, if parents were prepared to walk the short distance.
"Residents in our district often expect to be able to park right next to where they are going and would prefer not to have to walk from the car to their destination," council district living group manager Paul Dell said.
He said one of the main issues at Kensington was a shortage of fields for winter sport..
The council met with codes using the park at the start of the season asking them to advise parents and others that parking was limited and people should look for spaces in surrounding streets. There are 440 spaces in the Kensington car park and 130 in Lupton Ave as well as others in surrounding streets.
"Kensington is particularly challenging as the land around the park is all residential and would be expensive to purchase [land] to provide more carparks. Ongoing parking needs at Kensington will be assessed during a review of the Kensington Park management plan, being initiated this year," Mr Dell said.
"As part of the Long Term Plan, council is looking to develop a new urban park to address the district-wide issue of shortages of sports fields and take pressure off Kensington Park."
He said the council would continue to enforce parking regulations in this area where the parking of vehicles caused a danger to the public, such as parking on the grass verges along SH1, parking that blocked lines of sight and parking on yellow lines. Vehicles blocking residents' driveways will be towed away.
"I went up to Kensington on a Saturday morning and saw plenty of empty parking spaces just a few minutes walk away in Lupton Ave. People may not be able to park right on the door step but there is no excuse for parking illegally rather than walking for a few minutes," Mr Dell said.