The closest I've found is the episode Shore Leave, in which all the crew of the Enterprise are arguably part of landing parties:
On stardate 3025.3, the Federation starship USS Enterprise, under the command of Captain James T. Kirk, arrives at a planet in the Omicron Delta system. Scans reveal the planet to be congenial, and Kirk announces shore leave for all off-duty personnel.
Not long after beaming down, the landing parties experience strange occurrences.
In the novel Star Trek: Crisis on Centaurus, there's a clearer instance of Kirk sending down two landing parties:
With the Enterprise transporters out, Kirk decides, despite the risk involved, to send two landing parties to the planet in shuttlecraft. A party led by Spock will travel to the Defense Center to get its computers to shut down, while a party led by Kirk will go to McIverton to meet with the interim government.