We present a fully parallel nonlinearly implicit algorithm for the numerical simulation of some branching blood flow problems, which require efficient and robust solver technologies in order to handle the high nonlinearity and the complex geometry. Parallel processing is necessary because of the large number of mesh points needed to accurately discretize the system of differential equations. In this paper we introduce a parallel Newton-Krylov-Schwarz based implicit method, and software for distributed memory parallel computers, for solving the nonlinear algebraic systems arising from a Q2-Q1 finite element discretization of the incompressible Navier-Stokes equations that we use to model the blood flow in the left anterior descending coronary artery.