Heat the olive oil in a large pot or Dutch oven over medium-high heat. Add the onion, season with 1/2 teaspoon salt and cook, stirring occasionally, until soft and slightly browned, about 5 minutes. Add the chorizo and cook, stirring occasionally, until lightly browned around the edges, about 3 minutes. Add the garlic and cook, stirring, 1 minute. Add the potatoes, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until well coated, about 3 more minutes.
Add the chicken broth, bay leaves and paprika. Bring to a simmer, then reduce the heat to medium and cook 12 minutes. Add the chard; cover and cook, stirring, until the potatoes are tender and the chard is wilted, 3 more minutes. Season with salt and pepper. Ladle into bowls; top with parsley, drizzle with olive oil and serve with bread.