Place the butter in a small pot over medium heat. Melt the butter and toast the pasta 2 to 3 minutes or until golden in color, add the rice and stir to coat. Add 1 cup stock to the pot and bring the mixture to a bubble, reduce the heat to low, cover the pot and cook 15 minutes. Stir in parsley; turn off the heat and cover. Let stand to cool. Fluff the rice with a fork then place in container and store in the refrigerator.
Meanwhile, place the rinsed trimmed chicken pieces in a shallow pan with the onion and bay leaves and cover with just enough water to come up to the top of the chicken, bring the mixture to a boil and reduce the heat to a simmer and gently poach to cook through, about 12 to 15 minutes. Place the chicken in a bowl and shred the meat with 2 forks. Reserve the stock.
While the chicken poaches start the soup by heating a Dutch oven over medium-high heat with a turn of the pan extra-virgin olive oil, add the chorizo and brown a couple of minutes. Add the onions, carrot, celery, chile, garlic, and piquillos to the pot and season with a little salt and some black pepper. Cook for 7 to 8 minutes to soften, then stir in the saffron threads and paprika to combine. Strain the chicken cooking liquids, about 2 cups, into the pot then add 3 cups stock-in-a-box, chicken, chorizo, and tomatoes. Bring the soup to a bubble, simmer a few minutes to combine flavors then cool completely, cover and place in the refrigerator for make-ahead meal.
To reheat and serve, place the soup over medium to medium-high heat covered, stirring occasionally until heated through. Add the rice to the soup and return the soup to a bubble. Serve in shallow bowls.