Add half the parsley and remaining salt to the pot. Stir well. Place the cod fillet on top of the barley and vegetables in the pot. Cover and cook for about 6 to 7 minutes.
Use tongs to remove the cod from the pot, transferring to a plate. Use a fork to flake the cod and place back in the pot with remaining parsley and celery leaves. Stir well with a wooden spoon and add more water if needed. Simmer for about 2 to 3 minutes.
Divide soup into serving bowls and grate on Parmesan cheese.