Place cod, milk and bay leaves in a saucepan over medium heat. Bring to the boil. Reduce heat and simmer gently for 5 minutes. Drain, discard bay leaves and reserve the milk. Transfer cod to a bowl and set aside to cool. Clean the pan.
Thinly slice parsnips into rounds. Place in pan, cover with cold water and bring to the boil over medium heat. Cook for 5 minutes or until just softened. Drain and transfer to a bowl. Set aside.
Melt 80g butter in same pan over medium heat. Add onion, celery and carrot. Cook for 5 minutes or until softened. Stir in flour and cook for 2 minutes. Gradually add reserved milk, stirring until smooth. Cook for 3 minutes or until thickened. Set aside.
Flake cod. Add to sauce with egg and parsley, then season. Transfer to a 6-cup capacity pie dish. Place on a baking tray.
Melt remaining butter, add to parsnip and stir to coat. Layer over pie. Top with cheese. Bake for 30 minutes or until golden.