Not sure about private parks as we usually stay in the Provincial parks.
Here is the link for the Ontario Provincial Parks website. You can find all the parks available and make reservations (5 months to the day in advance).
The closest provincial park is Fairbanks. We stayed there a few years ago. It was nice but still a fair drive (57kms but close to an hour drive) from Sudbury with no gas nearby.
If you’re into birding and geology then the Fielding bird sanctuary, Big Nickel & Dynamic Earth (science north) tours are a must. Easy info if you google them.