Pour 1 1/2 cups water into a multi-functional pressure cooker (such as Instant Pot). Insert the trivet. Remove tough outer leaves and core from cabbage. Place cabbage, core-side down, on the trivet.

Close and lock the lid; set the vent to Sealing. Select Manual function according to manufacturer's instructions; set the timer for 9 minutes. Allow 10 to 15 minutes for pressure to build.

While cabbage is cooking, place rice in a small bowl and cover with water; set aside to soak. Measure out 1/3 cup tomato sauce (from 1 can) and pour into a large bowl. Add beef, pork, veal, onion, garlic, 1 teaspoon salt, parsley, and pepper; set aside.

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 the pot insert. Add remaining tomato sauce from the open can and 1 teaspoon salt to the pot. Replace the trivet in the pot.

Drain rice and add to meat mixture; stir to combine.

Separate cooked cabbage leaves carefully and shave off 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 remaining 1 can tomato sauce over rolls.

Close and lock the lid. Select the Stew/Meat function according to manufacturer's instructions; set the 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, about 10 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.