This paper develops a geometric method to estimate the error space of 3-DOF planar mechanisms with the Minimum Volume Ellipsoid Enclosing (MVEE) approach. Both the joint clearances and actuator errors are considered in this method. Three typical planar parallel mechanisms are used to demonstrate. Error spaces of their serial limbs are analyzed. Thereafter, limb-error-space-constrained mobility of the manipulator, namely, the manipulator error space is analyzed. The MVEE method has been applied to simplify the constraint modeling. A closed-form expression for the manipulator error space is derived. The volume of the manipulator error space is numerically estimated. The approach in this paper is to develop a geometric error analysis method of parallel mechanisms with clear algebraic expressions. Moreover, no forward kinematics computations have been performed in the proposed method, in contrast to the widely used interval analysis method. Although the estimated error space is larger than the actual one, because the enclosing ellipses enlarge the regions of limb error space, the method has an attractive advantage of high computational efficiency.