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The trajectory tracking control of parallel robots is challenging due to their complicated dynamics and kinematics. This paper proposes a position-based visual servoing (PBVS) approach for a 6-Revolute-Spherical-Spherical (6-RSS) parallel robot using adaptive sliding mode control in Cartesian space. A photogrammetry sensor C-Track 780 in the eye-to-hand configuration is adopted to measure the real-time pose of the robot end-effector, which can avoid the calculation of robot forward kinematics and provide more flexibility for controller design. An adaptive Kalman filter is utilized to deal with uncertain noises in visual measurements to increase the pose estimation accuracy. A sliding mode controller with strong robustness is designed to cope with system uncertainties, and a radial basis function (RBF) neural network is incorporated to realize the auto-tuning of the control gains, which make the robot effectively track different trajectories with time-varying conditions in real applications. Based on Lyapunov theorem, the stability analysis of the controller has been done. Experiments have been conducted to validate the effectiveness of the proposed strategy and illustrate the superiority of the designed controller.