Residents have reported a number of incidents where bins and areas around them have been damaged at Tapton Park.
Derbyshire police said officers were working with firefighters to prevent further incidents.
Resident Mark Jones, who uses the park at least once every day, has called for more patrols to crack down on the problem.
He said: “Vandalism is on the increase within the park in general, with fences and planting beds damaged in the past.
“It makes me feel disappointed that there is an element within the community here in Chesterfield who believe this is acceptable.
“I would like to see more police patrols in the park to show a presence.
"The park is one of the best we have in the town, it’s used by many people and we should try to ensure it remains that way.”
Mark says arsonists have targeted a number of different areas in the park, including close to the Innovation Centre car park, alongside the golf course and at the back of Tapton House.
A Derbyshire police spokesperson said: “We have received a report of bins and woodland at Tapton Park, off Brimington Road, which are believed to have been set on fire sometime overnight between Friday, April 22 and Saturday, April 23.
“Enquiries are ongoing and we would like to hear from anyone who may have been in the park and noticed any suspicious activity, or with information about the incident.
“Officers from our local Safer Neighbourhood policing team are working with the community safety team at Derbyshire Fire & Rescue Service to carry a walkabout in the area and consider steps that can be taken to help prevent further incidents, should it become a longer-term issue.”
Tapton House itself was the victim of vandalism when graffiti was spray-painted on its Georgian brickwork recently.
Tapton Park, Tapton House and its gardens were presented to Chesterfield Borough by Charles Paxton Markham in 1925.
Anyone with any information should contact Derbyshire police, quoting reference 22000233155:
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Phone – call 101.