Half of the enlisted Navy crew have their rank on their left sleeves (correctly), half of the crew have their rank on their right sleeves (incorrectly). Perhaps it was a joke that ran from the TV show? Actually, From 1841 until 1949 Boatswains Mates, Turret Captains, Signalmen, Gunners Mates, Fire Controlmen, Quartermasters, Mineman, and Torpedomen wore their rating/rank on their right sleeve.
The boat used to portray the PT-73 is much too small.
Boxes on aircraft are marked "USAF". There was no USAF during WWII. They would have either been marked USAAF or AAF.
The C-47 transport that Parker arrives in has red stripes through the white bars on the US national insignia, which were not added until 1948, 3 years after the end of World War II. (Other aircraft on the same airfield have the correct national insignia for late World War II.)
The world map on the wall in Harkness office are showing the borders as they were in the 1950*s.
When Binghamton and Parker are dangling from the C-47 in a jeep, stabilizing wires are clearly apparent: both in a close shot and in a long-shot of a model of the jeep.