Understanding vector fields resulting from large scientific simulations is an important and often difficult task. Streamlines, curves that are tangential to a vector field at each point, are a powerful visualization method in this context. Application of streamline-based visualization to very large vector field data represents a significant challenge due to the non-local and data-dependent nature of streamline computation, and requires careful balancing of computational demands placed on I/O, memory, communication, and processors. In this paper we review two parallelization approaches based on established parallelization paradigms (static de- composition and on-demand loading) and present a novel hybrid algorithm for computing streamlines. Our algorithm is aimed at good scalability and performance across the widely varying computational characteristics of streamline- based problems. We perform performance and scalability studies of all three algorithms on a number of prototypical application problems and demonstrate that our hybrid scheme is able to perform well in different settings.