This study proposes a computational and mathematical framework aimed at assessing the reliability of structural components within Floating Offshore Wind Turbines (FOWT) that reflects the various sources of uncertainties coupled between structural analyses, hydrodynamics, and aerodynamics. The limit state functions are represented through structural capacity and environmental demand models for selected structural failure modes that incorporate fully coupled aero-hydro-servo-elastic analysis. The fragility surfaces are developed for a selected benchmark wind turbine for both operating and parking conditions. The fragilities are also estimated under 50-year and 100-year environmental conditions in the selected U.S. coastal regions. It is found that the wind speed variations largely affect the fragility during non-operation, while wave height variations are significant during operation. Increased uncertainties in environmental parameters raised failure probabilities, especially in lower fragility ranges targeted by design codes. Analyses in U.S. coastal environments show both parking and operating conditions can be critical, challenging the previous focus on parking. Sensitivity studies reveal that under mild conditions, structural reliability is influenced by moment of inertia and material strength, but as environmental loads increase, these parameters become equally significant. Increased uncertainties in parameters lead to higher failure risks, especially below 25 m/s wind speeds.