In a large pot or French oven, add the olive oil and heat to medium. Add the chopped onion, bell pepper, celery and brown rice. Sauté, stirring frequently until vegetables have softened but are still al dente, about 10 minutes.
Add the chopped chicken and andouille sausage and continue sautéing until chicken is slightly browned, but not cooked through, about 3 minutes.
Add the chicken broth, white wine, diced tomatoes, Cajun seasoning, and dried thyme and stir everything together. Bring pot to a full boil, then reduce the heat and simmer for 40 to 50 minutes until rice is soft.
Add the peeled raw shrimp and fresh parsley and cook until shrimp is cooked through, about 5 minutes.
Allow the jambalaya to sit about 10 to 15 minutes before serving, allowing it will thicken up. Add salt as desired.