Letterboxes are permitted but must be registered. Online registration can be found at http://www.greatparks.org/secure/Geocache.htm. The following is copied from the registration page:

Guidelines For Placing Geocaches or letterboxes within the Hamilton County Park District:

• Must use a normal type of container. (i.e., it cannot resemble a pipe bomb or otherwise cause alarm to the public, etc.) and provide a description of it.
• Must be placed on or near an authorized trail. An authorized trail is one that is maintained by the HCPD’s Trail Crew, usually has either asphalt, crushed gravel or wood chip base. The container cannot be placed more than 25 feet from the trail. The container cannot be placed in any unsafe areas (ie, near a steep embankment).
• Placement or access to the container cannot damage park property or vegetation.
• All materials must be safe and appropriate for families and children.
• Requesting party is responsible for checking the container contents, as well as it’s proper location.
• The HCPD must be given the actual location of each container. (For example, The container is .5 miles into the Oakleaf Trail, under the log to the right of the trail sign that is titled “What Good is Dead Wood?”).
• Each request will be given a registration number by the HCPD that must be visible on the outside of each container.
• The HCPD will record the name, address, phone number and if applicable, the e-mail address of the requesting party associated with the container.
• Annual renewal is required for all active boxes.
• The requesting party must abide by all Park District By-Laws.

Letterboxing is allowed in the Geauga Park District, but you must follow a permit system. They tend to use the term 'geocaching' rather than 'letterboxing' in their materials, but the rules are the same for both hobbies.

The park authorities there also created an account on Atlas Quest, but it's doubtful they check it regularly. If you want to talk to someone about planting or finding letterboxes, you should probably contact them through their website or at the parks themselves.

The Geauga Park District includes the following parks:

Bass Lake Preserve
11445 Lakeview Drive, Munson Twp.

Beartown Lakes Reservation
18870 Quinn Road, Auburn/Bainbridge Twps.

Bessie Benner Metzenbaum Park
7940 Cedar Road, Chester Twp.

Big Creek Park
9160 Robinson Road, Chardon Twp.

Burton Wetlands Nature Preserve
15681 Old Rider Road, Burton/Newbury Twps.

Chickagami Park
17957 Tavern Road, Parkman Township

Eldon Russell Park
16315 Rapids Road, Troy Twp.

Frohring Meadows
16780 Savage Road, Bainbridge Twp.

Headwaters Park
13365 Old State Road (Route 608), Huntsburg/Claridon Twps.

The Maple Highlands Trail
Big Creek Spur
9280 Ravenna Road, Chardon Twp.

Mountain Run Station
on The Maple Highlands Trail
12601 Chardon Windsor Road, Hambden Twp.

Observatory Park
10610 Clay Street, Montville Twp.

Orchard Hills Park
11340 Caves Road, Chester Twp.

Sunnybrook Preserve
12474 Heath Road, Chester Twp.

Swine Creek Reservation
16004 Hayes Road, Middlefield/Parkman Twps.

The Rookery
10110 Cedar Road, Munson Twp.

The West Woods
9465 Kinsman Road (Route 87), Russell Twp.

The West Woods - Affelder House
15139 Chillicothe Road, Russell Twp.

Walter C. Best Wildlife Preserve
11620 Ravenna Road, Munson Twp.

Whitlam Woods
12500 Pearl Road, Hambden Twp.

Letterboxing is not allowed here, per orders of Mark J. Bollinger, the general manager.