The complex, nonintuitive kinematics of concentric tube robots (CTRs) can make their telemanipulation challenging. Collaborative control schemes that guide the operating clinician via repulsive and attractive force feedback based on intraoperative path planning can simplify this task. Computationally efficient algorithms, however, are required to perform rapid path planning and solve the inverse kinematics of the robot at interactive rates. Until now, ensuring stable and collision free robot configurations required long periods of precomputation to establish kinematic look-up tables. This article presents a high-performance robot kinematics software architecture, which is used together with a multinode computational framework to rapidly calculate dense path plans for safe telemanipulation of unstable CTRs. The proposed software architecture enables on-the-fly, incremental, inverse-kinematics estimation at interactive rates, and it is tailored to modern computing architectures with efficient multicore central processing units (CPUs). The effectiveness of the architecture is quantified with computational-complexity metrics and a clinically demanding simulation inspired from neurosurgery for hydrocephalus treatment. By achieving real-time path planning, active constraints (ACs) can get generated on the fly and support the operator in faster and more reliable execution of telemanipulation tasks.