Saute mushrooms and green onions in butter in a large skillet until the vegetables are tender. Add flour, stirring well. Cook for 1 minute, stirring constantly. Gradually stir in the milk.
Add parsley, salt, grated lemon rind, and pepper. Bring to a boil.
Arrange the fish fillets in a single layer over the sauce in the skillet; carefully spoon the sauce over the fillets. Cover, reduce heat, and simmer 12 to 15 minutes or until the fish flakes easily when tested with a fork (depends on the thickness of your fillets).