Parallel continuum robots have the potential to provide multi-degree-of-freedom articulation using a structure that is simple, compact, compliant, and highly scalable. These characteristics may be useful in micromanipulation, endoscopic robotic-assisted surgery, and human-robot interaction. Our prior work formulated a kinematic model which treats a parallel continuum robot as a set of multiple Cosserat rods with coupled boundary conditions. In this paper, we detail methods for the efficient numerical solution of this model at rates that enable real-time interactive simulation, motion planning, design optimization, and control. Exploitation of the model structure enables a significant reduction in the number of integrations required to evaluate the boundary value Jacobian matrix used in a shooting method. Our approach is used to teleoperate a prototype robot using real-time inverse kinematics solutions, and simulation tests show that inverse kinematics solutions are consistently computed at rates of several kilohertz using standard desktop computing hardware.