Heat oil in a pressure cooker over medium-high heat until shimmering. Add chorizo and cook, stirring, until just starting to crisp around the edges, about 2 minutes. Add onions and cook, stirring, until softened, about 3 minutes. Add paprika, chickpeas, tomatoes, chicken pieces, and broth. Season gently with salt and pepper.
Seal pressure cooker and bring to high pressure. Cook for 15 minutes. Cool pressure cooker under a cold running tap (if using an electric cooker, use the quick release valve), and open. Return to high heat and cook, stirring gently, until it achieves a thick, stew-like consistency, about 5 minutes. Stir in vinegar and season to taste with salt and pepper. Stir in parsley and serve, drizzling with more extra-virgin olive oil at the table.