Preheat oven to 350°F.
In a 12-inch cast iron skillet (or other oven-proof skillet), heat oil over medium-high heat. Add the sausage rounds and cook until browned, about 5 minutes. Use a slotted spoon to transfer the sausage to a plate, leaving the drippings in the pan.
Melt butter in the same skillet over medium-high heat. Add onion, celery, and bell pepper and cook, stirring, until the vegetables soften, 3 to 4 minutes. Add the garlic and cook, stirring constantly, until fragrant, about 30 seconds. Stir in the tomato paste, kosher salt, sugar, and pepper. Cook for 1 more minute.
Add the sausage back to the skillet, along with the water and rice. Bring to a simmer, then reduce the heat to low. Cook, uncovered, stirring occasionally, until the liquid has reduced slightly, about 7-9 minutes. Use a wooden spoon to scrape up any browned bits from the bottom of the skillet as you stir.
If you're not using an oven-proof skillet, transfer the mixture to a greased 2-3-quart baking dish. Cover the skillet (or baking dish) and bake until the rice is tender, about 30-40 minutes, stirring after the first 15 minutes. Fluff the rice, garnish with fresh parsley, and serve.