To enhance the cable vibration control performance of traditional passive linear viscous dampers（LVD），this paper investigates the cable vibration control performance of a negative stiffness parallel nonlinear viscous damper（NSNVD），which combines passive negative stiffness technique and nonlinear viscous damping characteristics. The linear equivalent model of the NSNVD is firstly obtained through the energy equivalent linearization method. Next，based on the complex modal analysis，the asymptotic and iterative solutions of the supplemental modal damping ratio of a taut cable are obtained，and compared with the numerical simulation solutions considering or neglecting the cable sag and bending stiffness. Accordingly，the applicability of the energy equivalent linearization method for the cable-NSNVD system is verified. Finally，based on the numerical simulation solutions considering the cable sag and bending stiffness，the parameter analysis of the single-mode vibration mitigation effect of the NSNVD and the optimization of the multi-mode vibration mitigation effect of the NSNVD on stay cables are carried out，which help to analyze the effects of the negative stiffness coefficient and viscous damping exponent of the NSNVD on single-mode and multi-mode vibration control performance of the cable. The results show that the passive negative stiffness of the NSNVD can significantly improve the single-mode vibration mitigation effect of stay cables. Moreover，both the negative stiffness and nonlinear viscous damping characteristics of the NSNVD are helpful to improve the multi-mode vibration mitigation effect of stay cables. Compared with the NVD and the NSLVD，the NSNVD presents more advantages in cable vibration control.