The abdominal aorta is the largest artery in the abdominal cavity that supplies blood flows to vital organs through the complex visceral arterial branches, including the celiac trunk (the liver, stomach, spleen, etc.), the renal arteries (the kidneys) and the superior and inferior mesenteric arteries (the small and large intestine, pancreas, etc.). An accurate simulation of blood flows in this network of arteries is important for the understanding of the hemodynamics in various organs of healthy and diseased patients, but the computational cost is very high. As a result, most researchers choose to focus on a portion of the artery or use a low-dimensional approximation of the artery. In the present work, we introduce a parallel algorithm for the modeling of pulsatile flows in the abdominal aorta with branches to the primary organs, and an organ-based two-level method for calculating the resistances for the outflow boundary conditions. With this highly parallel approach, the simulation of the blood flow for a cardiac cycle of the anatomically detailed aorta can be obtained within a few hours, and the blood distribution to organs including liver, spleen and kidneys are also computed with certain accuracy. Moreover, we discuss the significant hemodynamic differences resulted from the influence of the peripheral branches. In addition, we examine the accuracy of the results with respect to the mesh size and time-step size and show the high parallel scalability of the proposed algorithm with up to 3000 processor cores.