Heat a 12-in/30.5-cm skillet over medium-high heat and add the olive oil. When the oil shimmers, add the chicken thighs and brown them, about 3 minutes per side. Don’t try to turn the chicken if it’s stuck to the bottom of the pan; it will release once it is sufficiently browned. Transfer the chicken to a plate. (It will not be cooked through at this point.)
Add the sausage to the hot skillet, still over medium-high heat, and cook, stirring, until nicely browned on all sides, about 1 minute. Add the onion, bell pepper, garlic, Cajun seasoning, 1/4 tsp salt, and a few grinds of pepper and sauté the vegetables until they begin to soften, about 2 minutes longer. Add the rice and sauté until the grains begin to turn opaque, another minute or so.
Add the tomatoes with their juices and the chicken broth and return the chicken thighs to the pan, nestling them down into the liquid. Bring the mixture to a boil. Cover the pan tightly, reduce the heat to low, and simmer until the rice is tender and the chicken is cooked through, about 15 minutes. Uncover the pan, tuck the shrimp into the mix, and sprinkle the parsley over. Re-cover and cook until the shrimp is cooked through and firm to the touch, about 3 minutes more. Taste and season the jambalaya with more salt and pepper, if desired.
Mound the jambalaya in warmed shallow bowls and serve hot. Pass the hot sauce at the table, if desired.
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