Kinematic reliability is an essential index that assesses the performance of the mechanism associating with uncertainties. This study proposes a novel approach to kinematic reliability analysis for planar parallel manipulators based on error propagation on plane motion groups and clipped Gaussian in terms of joint clearance, input uncertainty, and manufacturing imperfection. First, the linear relationship between the local pose distortion coming from the passive joint and that caused by other error sources, which are all represented by the exponential coordinate, are established by means of the Baker–Campbell–Hausdorff formula. Then, the second-order nonparametric formulas of error propagation on independent and dependent plane motion groups are derived in closed form for analytically determining the mean and covariance of the pose error distribution of the end-effector. On this basis, the kinematic reliability, i.e., the probability of the pose error within the specified safe region, is evaluated by a fast algorithm. Compared to the previous methods, the proposed approach has a significantly high precision for both cases with small and large errors under small and large safe bounds, which is also very efficient. Additionally, it is available for arbitrarily distributed errors and can analyze the kinematic reliability only regarding either position or orientation as well. Finally, the effectiveness and advantages of the proposed approach are verified by comparing with the Monte Carlo simulation method.