With the increasing number of urban cars, parking has become the primary problem that people face in daily life. Therefore, many scholars have studied the automatic parking system. In the existing research, most of the path planning methods use the combined path of arc and straight line. In this method, the path curvature is not continuous, which indirectly leads to the low accuracy of path tracking. The parking path designed using the fifth-order polynomial is continuous, but its curvature is too large to meet the steering constraints in some cases. In this paper, a continuous-curvature parking path is proposed. The parking path tracker based on Model Predictive Control (MPC) algorithm is designed under the constraints of the control accuracy and vehicle steering. Firstly, in order to make the curvature of the parking path continuous, this paper superimposes the fifth-order polynomial with the sigmoid function, and the curve obtained has the continuous and relatively small curvature. Therefore, the superposition curve is used as a parallel parking path while the superposition curve and its inverse function curve are combined to form a perpendicular parking path. The coefficients of the superposition curve are calculated according to the constraint condition, the parking start point and end point. Thus, the parking path is determined.. Secondly, the vehicle kinematics model is established and a parking path tracker based on Model Predictive Control (MPC) algorithm is designed. Finally, the co-simulation analysis is performed using CarSim and Simulink. The simulation results show that the parking path curvature designed in this paper is continuous and the parking path tracker has a good tracking effect. The lateral error and longitudinal error can be controlled in the centimeter scale and the heading angle error is no more than 3°.