Combine 2 tablespoons of the oil with the crushed garlic, cayenne, and paprika in a large bowl and toss well with the chicken pieces to coat. Cover and set aside to marinate for 15 minutes or up to 2 hours at room temperature or overnight in the refrigerator, tossing once or twice during this time. Bring to room temperature before cooking.
Spread the chicken out on a plate (discarding the garlic) and salt generously on both sides. Reserve.
Heat the remaining 2 tablespoons oil over medium-high heat in a 10- to 12-quart heavy pot. When the oil shimmers, add the sausage and cook until lightly browned on both sides, 2 to 3 minutes. Using a slotted spoon, remove to a small bowl, leaving the fat in the pot.
Working in batches, if necessary, add the chicken (if skin-on, put skin-side down), and cook for 2 minutes. Reduce the heat to medium and continue cooking until deep golden brown on the first side, 8 to 10 minutes. Turn and cook until lightly browned on the second side, about 6 minutes more. Transfer the chicken to a plate and reserve.
Spoon off all but 4 tablespoons of fat in the pot and add the onions, green pepper, and bay leaves. Using a spatula and the moisture given off by the vegetables, scrape up any browned bits stuck to the bottom of the pot. Cook, stirring occasionally, until the vegetables are very soft and golden at the edges, 15 to 20 minutes. Reduce the heat if they threaten to burn. Stir in the chopped garlic and the thyme and cook for 2 minutes more. Increase the heat to medium-high and stir in the rice. Cook, stirring regularly until the rice is lightly toasted, about 3 minutes. Add the stock, black pepper, and 1 teaspoon salt (a bit more if your broth is unsalted) and again scrape up any browned bits on the bottom of the pot. Add the reserved chicken and any liquid on the plate and arrange the reserved sausage between the chicken pieces. Bring to a simmer, cover tightly, and reduce the heat so the broth is gently simmering. Cook until the broth is almost completely absorbed and the rice is just tender, about 25 minutes.
Let the jambalaya stand off the heat, covered, for 5 to 10 minutes and then gently stir in the scallions and parsley. Serve, passing hot sauce at the table.