Cut the corn from the cob. (I use a Bundt pan to help contain the corn as it's cut off the cob.) Set aside.
Heat the oil in a large Dutch oven or large stockpot over medium-high heat. Add the onion and garlic and cook until the onion is tender, about 5 minutes.
Stir in the corn, potatoes, lima beans, barbecue sauce, and chicken broth and bring to a boil.
Stir in the pulled pork, chicken, tomatoes, and paprika. Reduce the heat to low and simmer, uncovered, for about 20 minutes or until the vegetables are tender. Divide the stew between serving bowls and sprinkle with parsley, if desired.