The Lexington Parks and Recreation in Kentucky has jurisdiction over Lexington and Fayette County parks, and they have an approval system for geocaches that allow applies to those planting letterboxes:

Send an email to cdowns@lfucg.com with Letterbox Request in the Subject Line.

Include the following information in the body of the email:

1. Name, address and phone number of the owner.
2. Include the name the letterbox will have.
3. Posted clues plus the exact description to finding the LB.
4. Name of the park where letterbox is to be hidden.
5. Include a description of the container (Tupperware, ammo box, and film canister).
6. If you include a parking suggestion include this in the request.
7. Include if the letterbox is existing (hidden prior to May 1, 2003 only) or new cache.
8. Include 1-2 sentences describing where the letterbox is hidden.

Do not include pictures, diagrams, figures etc. Requests that are submitted incomplete will be returned. Once approval has been granted, you will be given an approval number. Include this number at the bottom of your letterbox description when you post the clues.

If a letterbox needs to be moved due to environmental impact, non-letterboxers finding it, etc. please resubmit the new clues and a new description of where the letterbox has been re-hidden.

The LFUCG Parks and Recreation cannot assume any liability for any letterbox placement. Letterboxers place boxes in Parks property at their own risk. This includes damage, destruction, and loss occurring to any LB placement.

The following is the policy for hiding boxes in Lexington and Fayette County parks.

• Groups/individuals must stay on trails at Raven Run and McConnell Springs where hiking off trail can cause ecological damage.
• Groups/individuals must obey operating hours and park rules and must park in designated parking areas only.
• The owner of the box should provide an email of request to the recreation supervisor/naturalist listed.
• Letterbox owners who already have boxes in Lexington parks should send an email in the same manner as a new box.
• LFUCG will respond to the request and will issue an approval number.
• Park staff will remove all boxes hidden without approval.
• As per standard Code of Conduct, no burying of letterboxes is allowed anywhere in the LFUCG Park system.
• Physical letterboxes are not allowed in the following areas: Phoenix Park, Thoroughbred Park, Gratz Park, riding areas at Masterson (including cross-country courses, jump areas, dressage and lesson rings, paddock areas, and indoor arenas), Camp Kearney during the summer, golf courses, pool areas, community center grassy areas, model airplane facility at Haley Pike, anywhere there are gates locked/fences to keep visitors out, and on the surface of any playing fields.

Note: Triangle Park is not LFUCG property so permission would have to come from Lexington Center Corp.

References
https://www.atlasquest.com/boards/message.html?gMsgId=84935

Kentucky does require pre-approval and a permit before you can plant a letterbox in Kentucky State Parks

Kentucky Geocaching page (Also applies to letterboxing)
Instructions for Establishing a Letterbox Site
Letterboxing Application and Permit