Uncertainties widely existing in modeling parameters, such as link length, joint clearance, and rotation angle, have the serious impact on the motion performance of industrial robots. In this study, a reliability analysis method based on evidence theory is proposed to uniformly analyze the influence of epistemic uncertainty and their correlation in modeling parameters on the positioning accuracy of robotic end effector. For the epistemic uncertainty derived from the limited sample data of modeling parameters, a generalized evidence theory model based on parallelotope frame is developed, which can uniformly quantify epistemic uncertainty and correlation of modeling parameters using the evidence framework of discernment and joint focal elements with same parallelotope features. To overcome the contradiction between analysis efficiency and accuracy for industrial robot positioning with nonlinearity, an efficient space affine collocation method is further proposed based on dimension reduction decomposition. Under the parallelotope evidence theory model, this method can provide an accurate reliability analysis result at a lower computational cost. A six degrees-of-freedom industrial robot is showcased to demonstrate the effectiveness and advantages of the proposed method in positioning accuracy reliability analysis.