This paper proposes a new framework for 3D noncontact stereovision-based displacement measurement in conjunction with Phase-based Video Motion Magnification (PVMM) to capture in-plane and out-of-plane vibrations of stay cables. In the proposed method, cable vibrations are captured by a pair of synchronized digital cameras, then PVMM method is applied to magnify micro vibrations of the cable (ambient responses) recorded by video cameras. Next, a camera calibration technique is proposed to convert the 2D image coordinates to 3D global coordinates, followed by the stereo calibration between the two cameras to synchronize the target points. Afterwards, a Centroid-based Bounding-Box Tracking (CBBT) technique is proposed and used to track the movements of the cable in each video frame. As a result, time-history displacements of the cable are extracted from the magnified videos. Finally, natural frequencies and damping ratios of the cable are extracted from the in-plane and out-of-plane displacements using Wavelet-screening Random Decrement Technique (WS-RDT). The proposed methodology has been validated through a couple of wind-tunnel tests of cable vibrations. The results have demonstrated a good agreement between the proposed method and conventional accelerometer-based measurement results.