In a large Dutch oven, melt butter over medium heat. Add the onions, celery, and bell pepper, and sauté until the vegetables are tender, about 5 minutes.
Stir in the garlic and flour, and cook for another 2 minutes, stirring constantly.
Gradually whisk in seafood stock, making sure to dissolve any flour lumps. Bring the mixture to a boil, then reduce heat to a simmer.
Add crawfish tails, cajun seasoning, salt, and pepper. Simmer for 45 minutes, stirring occasionally.
Stir in the heavy cream and parsley, and simmer for an additional 10 minutes.
Taste and adjust seasonings if necessary. Serve hot.