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Partial dynamic reconfiguration, often called run-time reconfiguration (RTR), is a key feature in modern reconfigurable platforms. In this paper, we present parallelism granularity selection (PARLGRAN), an application mapping approach that maximizes performance of application task chains on architectures with such capability. PARLGRAN essentially selects a suitable granularity of data-parallelism for individual data parallel tasks while considering key issues such as significant reconfiguration overhead and placement constraints. It integrates granularity selection very effectively in a joint scheduling and placement formulation, necessary due to constraints imposed by partial RTR. As a key step to validating PARLGRAN, we additionally present an exact strategy (integer linear programming formulation). We demonstrate that PARLGRAN generates high-quality schedules with: (1) a set of small test cases where we compare our results with the exact strategy; (2) a very large set of synthetic experiments with over a thousand data-points where we compare it with a simpler strategy that tries to statically maximize data-parallelism, i.e., only considers resource availability; and (3) a detailed application case study of JPEG encoding. The application case-study confirms that blindly maximizing data-parallelism can result in schedules even worse than that generated by a simple (but RTR-aware) approach oblivious to data-parallelism. Last, but very important, we demonstrate that our approach is well-suited for true on-demand computing with detailed execution time estimates on a typical embedded processor. Heuristic execution time is comparable to task execution time, i.e., it is feasible to integrate PARLGRAN in a run-time scheduler for dynamically reconfigurable architectures.