Missouri State Parks do allow letterboxing as long as you get permission first. An official permit can be found at:
MO State Park Permit Application (PDF file)
Missouri State Parks Website       
Missouri State Parks Policy: (PDF file)

Yes, the MDC allows letterboxing and geocaching with a Special Use Permit. See Geocaching and Letterboxing Permitted April 30, 2009 for more information.

Yes, the Greene County Park Board allows letterboxing and geocaching with a Special Use Permit. See Geocaching and Letterboxing Policy Guidelines for more information.

Allowed with prior approval. Contact Nancy Feakes, Recreation Manager, Mark Twain National Forest, 573-364-4621 x472 (2007).

Additionally, some municipalities in Missouri have began to develop their own policies for placement on park property.

Allowed without permit as per I Dig Toasters who spoke with park staff (2010).

Policy
Permit Application

Allowed with no permit (2007). Crestwood even sponsored their own geocache! Crestwood Parks

Jefferson City Letterboxing/Geocaching Policy

Allowed as per MOUR4Me who spoke with park staff (2010). However, beware of vandalism in their busy parks. Recommend using AQ find count restriction feature.

Allowed as per a letterboxer who spoke with park staff (2007). Parks website.

Allowed as per a letterboxer who spoke with park staff (2007).

St. Charles City Parks: allowed with permit. The cache containers must be clear and display the words "Official Geocache" or “Official Letterbox” clearly on the outside of the container. (2009)
Procedures

St. Charles City Parks: allowed with permit. The cache containers must be clear and display the words "Official Geocache" or “Official Letterbox” clearly on the outside of the container and they will assign a number that should also go on the label. (2009)
MO UR4Me helped them develop the permit, which they have never placed on their website, so it has been posted on Gateway Letterboxers: PERMIT.
They have not been able to keep up with the permits received; the Chief Ranger unofficially told MO UR4Me to put her boxes out and send in the permits, so she recommends doing the same. Then if you ever actually receive a permit number, you/someone can add it to the box.

St. Peters allows with prior permission of the parks board (2007). However, their current website says nothing about it. St. Peters Parks website

Gateway Arch and ALL NATIONAL PARKS (10/1/2009) - MIGHT be allowed; up to Superintendent of the park. See policy.

Allowed with no permit (2007). St. Louis County Parks allows letterboxes and geocaches. Parks website.

Allowed with no permit (2007). St. Louis City Parks allows letterboxes and geocaches. Parks website.

Allowed as per a letterboxer who spoke with park staff (2007).