Surgical manipulators are increasingly capable of approaching deep-seated pathologies through convoluted pathways due to the advances in the field of continuum robotics. This class of robots can be, in most cases, accurately modeled as a chain of cylindrical shapes. In order to safely and seamlessly telemanipulate these robots, which have complex nonintuitive kinematics, haptic guidance schemes have been developed that rely on accurate proximity queries (PQ) to calculate the distance between the continuum robot and the anatomy. This letter introduces an approach to accurately model the continuum robots using cylindrical-shaped segments with spherical or flat caps and then efficiently calculate the shortest distance to a triangle mesh. Implementations of efficient, analytical narrow phase PQ calculations for simple and complex geometrical primitives suitable for parallel computing hardware are presented together with an experimental validation which show improved performance suitable for real-time robotic applications. An in silico experiment comparing various root-finding algorithms, which are commonly used in PQ calculation or other optimization tasks are compared to assess their suitability when executed on different hardware. The implications of these experimental results are discussed, in particular with regards to the selection of a suitable proximity query algorithm depending on the available parallel computing hardware. Finally, an outlook of future improvements for dense dynamic anatomies is presented.