Comprehensive simulations of the entire biochemistry of cells have great potential to help physicians treat disease and help engineers design biological machines. But such simulations must model networks of millions of molecular species and reactions. The Stochastic Simulation Algorithm (SSA) is widely used for simulating biochemistry, especially systems with species populations small enough that discreteness and stochasticity play important roles. However, existing serial SSA methods are prohibitively slow for comprehensive networks, and existing parallel SSA methods, which use periodic synchronization, sacrifice accuracy. To enable fast, accurate, and scalable simulations of biochemistry, we present an exact parallel algorithm for SSA that partitions a biochemical network into many SSA processes that simulate in parallel. Our parallel SSA algorithm exactly coordinates the interactions among these SSA processes and the species state they share by structuring the algorithm as a parallel discrete event simulation (DES) application and using an optimistic parallel DES simulator to synchronize the interactions. We anticipate that our method will enable unprecedented biochemical simulations.