Put the fish in a deep frying pan with the bay leaf and 600ml/1 pint boiling water. Cover and simmer for 2 minutes. Turn off the heat and leave to stand, covered, for a further 5 minutes. Drain, reserving the liquid, then flake the fish.
Heat the oil and butter in a deep saucepan. Add the vegetables and garlic and fry over a high heat until starting to soften. Stir in the thyme and the reserved cooking liquid and bring to the boil. Reduce the heat and simmer for 10 minutes until the vegetables have softened.
Pour half the soup into a bowl and mash with a potato masher or fork. Return to the pan with the milk and fish. Simmer for 3 minutes, taste and season if needed. Sprinkle with chopped parsley to serve.