High-resolution blood flow simulations have potential for developing better understanding biophysical phenomena at the microscale, such as vasodilation, vasoconstriction and overall vascular resistance. To this end, we present a scalable platform for the simulation of red blood cell (RBC) flows through complex capillaries by modeling the physical system as a viscous fluid with immersed deformable particles. We describe a parallel boundary integral equation solver for general elliptic partial differential equations, which we apply to Stokes flow through blood vessels. We also detail a parallel collision avoiding algorithm to ensure RBCs and the blood vessel remain contact-free. We have scaled our code on Stampede2 at the Texas Advanced Computing Center up to 34,816 cores. Our largest simulation enforces a contact-free state between four billion surface elements and solves for three billion degrees of freedom on one million RBCs and a blood vessel composed from two million patches.