Heat the olive oil in a large pot over medium heat. Add the chicken pieces and cook until lightly browned on all sides, about 6 minutes.
Add the onion, garlic, celery, carrots, and parsnips to the pot. Cook for 5 minutes, stirring now and then, until the vegetables begin to soften.
Stir in the smoked paprika, salt, and black pepper. Cook for 1 minute to toast the spices.
Add the potatoes, chicken broth, and bay leaves. Bring the mixture to a boil.
Reduce the heat to low, cover the pot, and simmer for 25 to 30 minutes until the potatoes and other vegetables are tender and the chicken is cooked through.
Remove the bay leaves. Ladle the stew into bowls and top with chopped parsley before serving.