Put fish trimmings and water into a medium-size saucepan. Bring to a boil; reduce beat. Cover pan; simmer 20 minutes.
In a blender or food processor, process fish until it forms a smooth paste. Add 1 teaspoon chopped parsley. Season with salt and pepper. Roll 1/2 of paste in small balls 1/2 inch in diameter. Flatten out remaining paste in a rectangular cake 2 inches wide. Cut in strips. Deep-fry strips until crisp and golden. Drain on paper towels. Keep hot.
Strain fish broth through a fine sieve into a 2-quart saucepan. Bring back to a boil. Add fishballs. Cook gently 5 minutes.
Meanwhile, in a large pan of boiling salted water, cook noodles until just tender to the bite.
Drain; divide among 4 heated bowls. Divide fishballs among bowls. Pour in broth. Arrange fried fish strips on top. Garnish with green onions and additional chopped parsley.