In a heavy-bottomed pot, heat the oil over medium heat. Add the onion and garlic, and cook until translucent and soft, about 3 minutes. Add the thyme and cook until fragrant, about 2 minutes. Add carrots and celery and sweat until translucent, about 3 minutes. Add the flour and cook for 3 minutes, stirring constantly, until all the vegetables are coated. Slowly pour in the chicken or turkey stock, stirring continuously to break up any lumps of flour.
Add the turkey meat. Season with salt and pepper to taste. Bring to a boil, then reduce heat. Simmer for 20 minutes until the liquid has reduced a little and the soup has slightly thickened. Taste and adjust the seasoning if needed. Ladle into bowls, garnish with parsley, and serve immediately.