Combine the turnips, potatoes, carrots, garlic, bay leaf, thyme, a pinch each of salt and pepper, and 4 cups of water in a large saucepan. Bring to the boil over moderate heat, reduce to a simmer, cover and cook for 15 minutes or until tender. Drain the vegetables and garlic, reserving 1/2 cup of the cooking liquid. Remove and discard the bay leaf.
Mash the vegetables and garlic cloves with a potato masher, adding the reserved cooking liquid and the oil. Stir in the Parmesan just before serving.