Traffic assignment methods are some of the key approaches used to model flow patterns that arise in transportation networks. Since static traffic assignment does not have a notion of time, it is not designed to represent temporal dynamics that arise as vehicles flow through the network and demand varies through the day. Dynamic traffic assignment methods attempt to resolve these issues, but require significant computational resources if modeling urban-scale regions (on the order of millions of links and vehicles) and often take days of compute time to complete. The focus of this work is two-fold: 1) to introduce a new traffic assignment approach - a quasi-dynamic traffic assignment (QDTA) model and 2) to describe how we parallelized the QDTA algorithms to leverage High-Performance Computing (HPC) and scale to large metropolitan areas while dramatically reducing compute time. We examine and compare different scenarios, including a baseline static traffic assignment (STA) and a quasi-dynamic scenario inspired by the user-equilibrium (UET). Results are presented for the San Francisco Bay Area which accounts for 19M trips/day and an urban road network of 1M links. We utilize an iterative gradient descent method, where the step size is selected using a Quasi-Newton method with parallelized cost function evaluations and compare it to using pre-defined step sizes (MSA). Using the parallelized line search provides a 16 percent reduction in total execution time due to a reduction in the number of gradient descent iterations required for convergence. The full day QDTA comprising 96 optimization steps over 15 minute intervals runs in about 4 minutes on 1,024 cores of the NERSC Cori computer, which represents a speedup of over 36x versus serial execution. To our knowledge, this compute time is significantly lower than other traffic assignment solutions for a problem of this scale.