Season the chicken and pork pieces liberally with salt (1 tsp), pepper (1 tsp), and smoked paprika (2 tbsp) all over.
Heat the oil in a large pot, preferably a Dutch oven, over medium-high heat. Working in batches, if necessary, add the chicken and pork to the pot and cook until nicely browned all over, about 11 minutes. Remove pieces with a slotted spoon onto a platter. Set aside.
Spoon out all but 2 tbsp of oil from the pot. Still over medium-high heat, add the onions and cook until soft, about 4 to 5 minutes. Add the garlic and cook for another minute. Add the thyme and parsley and cook for another minute. Add the tomato paste and stir with a wooden spoon until fully incorporated, about another 2 minutes.
Add the broth, tomatoes, corn, okra, lima beans, potatoes, Worcestershire sauce, cooked chicken, pork, 1½ tsp salt, and 1 tsp pepper. Stir to combine. Increase heat and bring to a boil.
Reduce heat and simmer the stew for 1 hour until potatoes are very soft. In the last few minutes of simmering, smash a few of the potato cubes against the side of the pot to help thicken the stew. Taste and add more salt, if desired. Serve at once.