The planetary roller screw mechanism (PRSM) is a novel precision transmission mechanism that realizes the conversion between linear and rotary motions. The contact characteristics of helical surfaces directly determine PRSM’s performance in load-carrying capacity and transmission accuracy. Therefore, studying the contact characteristics of PRSM forms the fundamental basis for enhancing its transmission performance. In this study, a three-dimensional parametric analysis method of contact characteristics is proposed based on the PRSM meshing principle and PyVista (a high-level API to the Visualization Toolkit). The proposed method considers the influence of machining errors among various thread teeth. The effects of key machining errors on contact positions and axial clearance, as well as their sensitivities, are analyzed. With excellent solution accuracy, this method exhibits higher calculation efficiency and stronger robustness than the analytical and numerical meshing models. The influence of nominal diameter and pitch errors of the screw, roller, and nut on the axial clearance follows a linear relationship, whereas flank angle errors have negligible effects on the axial clearance. The corresponding influence coefficients for these three machining errors on the axial clearance are 0.623, 0.341, and 0.036. The variations in contact positions caused by individual errors are axisymmetric. Flank angle errors and roller diameter errors result in linear displacements of the contact points, whereas pitch errors cause the contact points to move along the arc of the roller diameter. Based on the proposed three-dimensional parametric contact characteristics analysis method, the Fuzzy C-Means clustering algorithm considering error sensitivity is utilized to establish a component grouping technique in the selective assembly of critical PRSM components, ensuring the rational and consistent clearances based on the given component’s machining errors. This study provides effective guidance for analyzing contact characteristics and grouping in selective assembly for PRSM components. It also presents the proposed method’s potential applicability to similar calculation problems for contact positions and clearances in other transmission systems.