Heat the butter and oil in a saucepan set over a medium heat until hot. Add the onion and sweat with a little salt for 5 minutes until softened.
Add the parsnips and spices, and cook for 3 minutes, stirring occasionally. Add the stock, stir well, and cook until simmering.
Pour the contents of the pan into a slow cooker. Cover and cook on low for 5 hours until the parsnips are tender.
Puree the soup in a food processor or using an immersion blender until smooth. Stir in the milk and return the soup to the slow cooker, cooking on high for 15 minutes until hot. Season to taste.
Ladle into soup bowls and top with a swirl of cream and a sprinkle of chopped tarragon before serving.