The goal of this paper is the development of a mixed integer programming formulation for the production planning and scheduling of parallel (single-stage) continuous processes in the presence of sequence-dependent switchover times and costs for product families, and sequence-independent switchovers for products belonging to the same family. Our formulation combines a discrete-time planning (big-bucket) grid with a continuous-time treatment of the scheduling decisions within each period and across adjacent periods. At the production planning level, it handles product orders at intermediate due dates and accounts for holding and backlog costs. At the scheduling level, it accounts for equipment unit constraints, switchover times and costs, maintenance activities, and idle production periods. The proposed model was motivated from and implemented to a real-world production facility. It can effectively address industrial-scale planning-scheduling problems. Importantly, it yields solutions that are considerably better than the solutions currently obtained using commercial tools.