Coastal civil infrastructure can be susceptible to damages caused by hurricanes throughout its service life. Vulnerability assessment is a key component in hazard risk management of civil infrastructure systems. Previously, most studies assume storm parameters are independent when computing the vulnerability of infrastructure to hurricanes. Due to the complicated interactive effects between storm parameters during hurricanes, the independent model may mis-specify such intercorrelation, thus resulting in inaccurate estimation of the probability of failure. This paper proposes a copula-based vulnerability assessment framework to investigate the impact of dependent storm parameters on the vulnerability of civil infrastructure subjected to hurricanes. The developed framework is applied to a typical simply supported bridge to compute the probability with respect to deck unseating failure under hurricane hazards. The copula approach provides superior efficiency in modeling dependency between the maximum wave height and peak water level, by separately considering marginal distributions and the joint effects. Probabilistic wave-induced load acting on the bridge deck is computed using a three-dimensional computational fluid dynamics model, incorporating uncertainties associated with hazard parameters. The effect of correlated storm parameters and the tail dependence characteristics on the bridge vulnerability is investigated by using different copula models, including Clayton, Gaussian, and Hüsler-Reiss copula functions. By incorporating dependent hazard parameters, the failure probability of the bridge may be significantly increased for non-major hurricanes (e.g., with a 50-year return period), while the failure probability under major hurricanes (e.g., with a 500-year return period) may not be severely affected.