How to Make It
Gently prick the duck skin all over with a sharp paring knife (without piercing the meat) and place it on a rack set in the sink.
Bring a small pot of water to a boil. Slowly pour the water all over the duck to render some of the fat; tilt the duck to drain the cavity. Transfer the rack with the duck to a baking sheet; pat the duck dry with paper towels.
In a small bowl, combine the thyme, allspice, garlic powder, black pepper, crushed red pepper, onion powder and sugar. Rub the spice mix inside the cavity and all over the duck. Refrigerate uncovered for 1 to 2 days.
Preheat the oven to 300°. Season the duck with salt and stuff the cavity with half of the carrots and garlic. Transfer the duck to a rack (preferably V-shaped) set in a roasting pan and add the onion, chicken stock and the remaining carrots and garlic to the pan.
Roast the duck for about 1 hour, until lightly golden and an instant-read thermometer inserted in the thickest part of the thigh registers 140°. Increase the oven temperature to 450° and roast for 15 minutes longer, until the skin is deep golden and crispy and an instant-read thermometer inserted in the thickest part of the thigh registers 160°. Let the duck rest for 20 minutes.
Tilt the duck to release the juices into the roasting pan, then strain the pan juices into a medium saucepan and degrease with a spoon. Bring to a boil. Whisk in the butter and simmer for 5 minutes, until slightly thickened. Stir in the parsley and season with salt and pepper. Carve the duck and serve with the pan sauce.