In a medium pot, heat 1 tablespoon oil over medium-high. Add celery, bell peppers, and onion and season with salt and pepper. Cook, stirring often, until vegetables soften, 5 minutes. Stir in rice and 1 1/2 cups water; bring to a boil. Cover, reduce to a simmer, and cook until rice is tender, about 15 minutes. Remove pot from heat and let stand, covered, 5 minutes. Fluff rice with a fork and stir in parsley.
Meanwhile, in a small skillet, heat 1 teaspoon oil over medium-high. Add sausages and cook until browned on all sides, about 10 minutes. Reduce heat to low, cover, and cook until sausages are cooked through, 8 to 10 minutes. Let rest 2 minutes before slicing crosswise.
In a medium bowl, toss spinach with vinegar and 2 teaspoons oil. Season to taste with salt and pepper; serve spinach alongside rice and sausages.