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Wildfires have been, recently more-frequently, posing significant threats to the safety and reliability of electric power delivery infrastructure, calling for the development of dedicated risk and vulnerability assessment frameworks. Fragility functions are essential tools for probabilistic risk assessment to estimate the failure likelihood of the system components as a function of hazard intensity. Widely exposed to the environment and atmospheric stressors, overhead power line conductors are among the most susceptible electric delivery infrastructure to approaching wildfires. Beyond the state-of-the-art fragility models, this article proposes novel parameterized wildfire fragility functions that can capture the impact of different environmental conditions and wildfire severity measures on overhead power line conductors. In doing so, wildfire behaviour is comprehensively characterized in order to model the temperature change in overhead conductors in the face of progressing wildfires. A novel parametric fragility model, which is a function of a set of physical and environmental features (e.g., conductor height, landscape slope, the dryness of fuel, fuel depth, wind speed, flame length), is then suggested to assess the failure likelihood of conductors given a permissible safety conductor temperature rise. A set of fire features (e.g., fire line intensity, fire rate of spread, and the angle of the fire approaching the conductors) is embedded in the proposed fragility model to capture the wildfire uncertainties. The numerical results reveal that, unlike the commonly-used fragility models, the proposed parameterized fragility function is able to accurately represent the impact of critical physical, environmental, and fire features in determining the vulnerability of power distribution lines when facing wildfire emergencies.