Robot-assisted surgical systems have been widely applied for minimally invasive needle biopsies thanks to their excellent accuracy and superior stability compared to manual surgical operations, which lead to possible fatigue and misoperation due to long procedures. Current needle biopsy robots are normally customed designed for specific application scenarios, and only position-level kinematics are derived, preventing advanced speed control or singularity analysis. As a step forward, this paper aims to design a universal needle biopsy robot platform which features 6 DoF 3-RRRS (Revolute–Revolute–Revolute–Spherical) parallel structure. The analytical solutions to its nonlinear kinematic problems, including forward kinematics, inverse kinematics, and differential kinematics are derived, allowing fast and accurate feedback control calculations. A multibody simulation platform and a first-generation prototype are established next to provide comprehensive verifications for the derived robotic model. Finally, simulated puncture experiments are carried out to illustrate the effectiveness of the proposed method.