In this paper, a comprehensive study on the parametric and auto-parametric instability of taut moorings due to combined hydrodynamic and ice loads is conducted. In this regard, a modal analysis is performed that includes the geometric nonlinearity of the mooring cables along with the combined effect of hydrodynamic and ice loads. The stability of the mooring cable around the 2:1 internal resonance region is analysed in the presence of out-of-plane ice-load which is calculated using a semi-empirical ice load-penetration evolution. It is shown that the geometric nonlinearity along with coupled hydrody namic and ice loads play an important role in the modal interaction which may lead to the large amplitude vibrations, i.e., instability. Stability boundaries near the 2:1 resonance region are determined through simulations by observing the onset of the oscillation in the out-of-plane modes. Beyond this boundary, the instability or jump in the response may occur and this could lead to the fatigue failure of the mooring cables. A parametric study on the stability boundary based on varying cable pretension and fairlead slope is also conducted. This reveals that increasing the cable pretension and the fairlead slope can effectively mitigate the parametric and auto-parametric instability in out-of-plane modes of mooring cables.