The nonlinear dynamics of a simplified nonlinear model of a floating offshore wind turbine under wind and sea excitations is investigated. The motion of sea waves is represented as the heave displacement of the turbine tower, leading to a parametric excitation of both the along-wind and cross-wind displacements. The study explores the occurrence of a 1:1 internal resonance between along-wind and cross-wind oscillations using an asymptotic approach grounded in the multiple scale method. The findings highlight a rich dynamical scenario, which is significantly influenced by the sea excitation. In particular, a detrimental contribution to the system stability can be pointed out for low sea amplitude values, while a stabilizing effect is displayed for moderate to high values of the sea excitation. A behavior chart is developed to illustrate the model response in relation to the forcing parameters.