Flexible stay cables on cable-stayed bridges are three-dimensional. They sag and flex in the complex wind environment, which is a different situation to ideal rigid cylinders in two-dimensional wind flow. Aerodynamic interference and the response characteristics of wake galloping of full-scale parallel cables are potentially different due to three-dimensional flows around cables. This study presents a comprehensive wind tunnel investigation of wake galloping of parallel stay cables using three-dimensional aeroelastic cable models. The wind tunnel study focuses on the large spacing instability range, addressing the effects of cable separation, wind yaw angle, and wind angle of attack on wake galloping response. To investigate the effectiveness of vibration suppression measures, wind tunnel studies on the transversely connected cable systems for two types of connections (flexibility and rigidity) at two positions (mid-span and quarter-span) were also conducted. This experimental study provides useful insights for better understanding the characteristics of wake galloping that will help in establishing a guideline for the wind-resistant design of the cable system on cable-stayed bridges.