There is an ever increasing rate of digital information available in the form of online data streams. In many application domains, high throughput processing of such data is a critical requirement for keeping up with the soaring input rates. Data stream processing is a computational paradigm that aims at addressing this challenge by processing data streams in an on-the-fly manner. In this thesis, we study the problem of automatically parallelizing data stream processing applications to improve throughput. The parallelization is automatic in the sense that stream programs are written sequentially by the application developers and are parallelized by the system. We adopt the asynchronous data flow model for our work, where operators often have dynamic selectivity and are stateful. We solve the problem of pipelined fission, in which the original sequential program is parallelized by taking advantage of both pipeline and data parallelism at the same time. Our solution supports partitioned stateful data parallelism with dynamic selectivity and is designed for shared-memory multi-core machines. We first develop a cost-based formulation to express pipelined fission as an optimization problem. The bruteforce solution of this problem takes a very long time for moderately sized stream programs. Accordingly, we develop a heuristic algorithm that can quickly, but approximately, solve this problem. We provide an extensive evaluation studying the performance of our solution, including simulations and experiments with an industrial-strength Data Stream Processing Systems (DSPS). Our results show good scalability for applications that contain sufficient parallelism, closeness to optimal performance for the algorithm.