In a large, heavy-bottomed Dutch oven, preferably cast iron or enamel cast iron,  heat the vegetable oil over medium-high heat until shimmering hot. When a pinch of flour blooms on the surface when added to the oil, scatter in the flour and stir quickly and thoroughly, combining the oil and flour evenly into a thick, smooth roux.  Continue cooking, stirring often, as the roux turns from pale yellow to a rich, deep brown, 20 to 35 minutes. Adjust the heat as needed to keep the roux darkening slowly, without bubbling up or burning. It should be darker than peanut butter, about the color of coffee with some cream, a deep brown, but not a dark brown.

Add the onion, celery and bell pepper to the Dutch oven and stir well, heating them up and coating them with the roux. Let them cook, stirring often, until the vegetables are fragrant, softened, shiny and evenly coated with the roux, 2  to 3 minutes.  Add the garlic, thyme, salt, pepper, paprika and cayenne; stir well and cook 2 more  minutes.

Add the chicken stock, stir well and bring it to a lively boil, stirring often to dissolve the roux into the stock.  When everything is boiling nicely, add the chicken pieces.  When the stock returns to a rolling boil, adjust the heat to maintain a lively simmer and cook, stirring now and then, until the chicken is cooked through and very tender, 45 minutes to 1 hour.

While you are waiting, begin to make your rice.  Also, chop the andouille into bite-size pieces.  Halve each link lengthwise and then cut it crosswise into half-moon slices about 1/4 inch thick.

Remove the chicken from the pot and set it out on a platter to cool.  As soon as it is cool enought to touch, pull the meat from the bones, discarding the bones and skin.  Tear or chop the chicken very coarsely and then return the meat to the pot.  Add the andouille and stir.  Cook, stirring now and then, until the sausage has seasoned the gumbo and softened, and eveything has come together into a rich, flavorful, slightly thickened stew, about 30 minutes more.

Remove the pot from the heat and stir in the green onions and parsley.  Serve the gumbo in bowls over rice or with the rice on the side, garnished with more green onions.