Arsonists have badly damaged play equipment at the Leeside playground, which recently benefited from £14,000 of investment.
The damage was done on the evening of Saturday 30 April and the Fire Service were called to extinguish the blaze.
Cllr Stephen Fenton has asked council officers to repair or replace the damaged equipment and surfacing.
It was reported to the police shortly after 12.50am on Tuesday 14 October 2014, by North Yorkshire Fire and Rescue Service, who were dealing with a car fire on Tudor Road, Acomb.
The suspect was seen placing something underneath a BMW before setting it on fire.
After the incident, the person responsible headed towards Hamilton Drive.
As a result of the fire, the BMW 3 series was completely destroyed and a Citroen Berlingo and a Mazda were damaged. The front of the house was also damaged by the fire.
Detective Sergeant Mark Jackson, of York CID, said: “The act of arson took place very close to the house where people were sleeping. This fire could easily have led to potentially devastating consequences.
“If you recognise the person captured on camera, or saw someone matching their description in the Tudor Road, Gale Lane or Middleton Road areas around the time of the fire, I urge you to contact the police immediately.
“I am also keen to speak to anyone who works at local garages who may have sold fuel to someone filling a petrol can, and ask them to get in touch straight away.”
Anyone who can assist with this investigation should contact York CID on 101, select option 2 and ask for the department by name. Alternatively, information can be emailed to yorkcid@northyorkshire.pnn.police.uk.
Or call Crimestoppers anonymously on 0800 555111, quoting reference number 12140177124.