A vector‐river network explicitly uses realistic geometries of river reaches and catchments for spatial discretization in a river model. This enables improving the accuracy of the physical properties of the modeled river system, compared to a gridded river network that has been used in Earth System Models. With a finer‐scale river network, resolving smaller‐scale river reaches, there is a need for efficient methods to route streamflow and its constituents throughout the river network. The purpose of this study is twofold: (1) develop a new method to decompose river networks into hydrologically independent tributary domains, where routing computations can be performed in parallel; and (2) perform global river routing simulations with two global river networks, with different scales, to examine the computational efficiency and the differences in discharge simulations at various temporal scales. The new parallelization method uses a hierarchical decomposition strategy, where each decomposed tributary is further decomposed into many sub‐tributary domains, enabling hybrid parallel computing. This parallelization scheme has excellent computational scaling for the global domain where it is straightforward to distribute computations across many independent river basins. However, parallel computing for a single large basin remains challenging. The global routing experiments show that the scale of the vector‐river network has less impact on the discharge simulations than the runoff input that is generated by the combination of land surface model and meteorological forcing. The scale of vector‐river networks needs to consider the scale of local hydrologic features such as lakes that are to be resolved in the network.