It happened between 12.20am and 12.30am yesterday when a vehicle was deliberately driven onto the grass at Adderstone Field, which is part of the forest near Pickering.
The large open field is promoted as the ideal place for visitors to have a barbecue or play balls games, and has adventure play equipment for children to enjoy.
A North Yorkshire Police spokesman said: "We are conducting enquiries and are requesting the public’s assistance to establish the full circumstances surrounding the incident.
"In particular, we are appealing for information about the vehicle that caused the damage."
Anyone with information is asked to email [email protected] or call 101, select option 2, and ask for Louise Neville-Beck.
Information can also be passed anonymously to Crimestoppers on 0800 555 111.
Please quote the North Yorkshire Police reference number 12170139399.