Cudworth Park, on Royston Road, has been targeted a number of times in the past, after being vandalised, covered in graffiti and now set alight.
Coun Joe Hayward, told the Chronicle an investigation will be taking place and the culprits will be named and shamed as a result.
“This climbing frame is used by children across the borough,” Joe said.
“Due to the financial constraints, it is unlikely the equipment will be replaced.
“I don’t know what these people have achieved from doing this - apart from depriving children from using it and spoiling their fun.
“The park has been vandalised a lot, and the council have always paid for the repairs, but I’d say that’s unlikely in the current climate.
“If it doesn’t get replaced, parents and carers will have the minority of a few people to blame.
“It’s just a shame a small group of people can ruin things for the majority.
“Any information that leads to these people being prosecuted will be given a reward from me to say thank you.”
Coun James Higginbottom, cabinet spokesperson for environment and highways, said: “It's enormously frustrating to hear that the playground in Cudworth has been subjected to yet more vandalism, including arson against children's play equipment.
"This disgraceful behaviour will not be tolerated in our borough.
“Those who are committing this vandalism will be found and they will be held to account.
“The vast majority who enjoy our parks and public spaces should not have to pay for the actions of a mindless few.
"We're working with South Yorkshire Police and South Yorkshire Fire and Rescue to identify the offender.
“We have had to remove the play equipment for safety reasons as the cost to replace the damage is not affordable at this time.