WWII military Jeeps (GP, or general-purpose vehicles) were equipped with solid-rubber tires, much like farming tractors were at the time. The two reasons for this were that Jeeps were often parachuted out of cargo planes (to keep the planes from having to land, making them enemy targets) and the tires had to survive the landing; the other reason was to preclude any need to change a flat tire while in use in open combat. Parker's pistol couldn't have flattened a solid-rubber tire.
When McHale's fake Germans start to chase Binghamton and Carpenter dressed as Germans one of their rifles is left behind leaning against the big rock.
Parker shoots his side arm and it has real ammo in it but it should have been loaded with blanks like all the other weapons.
When the pillow fight begins it's easy to see that Parker's pillow is actually just a pillow case filled with feathers.