Pour 1 1/2 cups water into a multi-functional pressure cooker (such as Instant Pot(R)). Insert the trivet.
Remove tough outer leaves and core from the cabbage. Place cabbage, core-side down, on the trivet.
Close and lock the lid, and set vent to sealing. Select Manual function according to manufacturer's instructions; set timer for 9 minutes. Allow 10 to 15 minutes for pressure to build.
While cabbage is cooking, combine rice and 1/2 cup water in a small bowl. Combine beef, pork, veal, 1/3 cup tomato sauce, onion, garlic, parsley, salt, and pepper in a separate bowl.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Transfer cabbage to a plate. Drain water from pot insert. Add remaining tomato sauce from the opened can and 1 teaspoon salt. Replace the trivet in the pot.
Drain rice and add to the bowl with the meat mixture.
Separate cooked cabbage leaves carefully and shave off the tough veins. Place a cabbage leaf on a work surface; fill with 2 tablespoons meat mixture and roll up, tucking in the sides. Place rolled leaf on the trivet in the pot. Continue with remaining leaves and filling, placing the first layer facing the same way and the second layer facing the opposite way. Pour the remaining can of tomato sauce over the rolls.
Close and lock the lid. Select the Stew/Meat function according to manufacturer's instructions; set timer for 22 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions for 10 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.
