Coun Steve Green, who represents the Monk Bretton ward, said he was walking his dog when he came across the damage on Littleworth Lane - which he believes was caused on either Friday or Saturday.
“I live on the estate behind so I walk my dog on there regularly,” he said.
“I almost cried when I saw what had happened.
“It must have happened on Friday or Saturday with the sole purpose of ruining the pitch.
“It’s going to cost a lot of money - there are parts of the pitch which now have no grass at all.”
Coun Green is hoping that a local resident will come forward with information about who owns or where the quad bikes are kept after seeing the state of the park.
“These people are not only riding along the park but they’re going up and down Lundwood,” he added.
“Hopefully when someone sees the disgraceful pictures they will come forward with some information.
“Somebody knows where this quad is kept.”
South Yorkshire Police’s off-road team confirmed they’re aware of the incident and have urged residents to get in touch.
“This is just mindless vandalism and will ruin sport for all that use the pitch,” a spokesperson added.
“If you have any information please get in touch.”