This paper presents an extended approach for computing unique solutions to forward and inverse kinematics of a three degrees-of-freedom spherical parallel manipulator (SPM) with coaxial input shafts and all revolute joints that has an unlimited rolling motion property. The approach is formulated in the form of easy-to-follow algorithms. Numerical and simulation case studies are conducted on a novel coaxial SPM design model demonstrating its multiple possible solutions of the forward and inverse kinematics problems constituting assembly and working modes of the manipulator, respectively. It is confirmed that the proposed approach allows computing of a unique solution corresponding to the specific assembly or working mode of a coaxial SPM. Furthermore, a 3D printed coaxial SPM prototype is presented in detail for experimental verification of the performed numerical and simulation analyses. The obtained results can be applied in the design of realtime orientation control systems of coaxial SPMs.