Parametric excitations can be encountered in a large variety of dynamic mechanical systems. Such excitations must be taken into account at the design stage in order to avoid the frequency operating ranges in which the mechanical systems may exhibit instabilities sources of failures. Consequently, systems with parametric excitations have been widely studied over the past decades and many relevant works have been published in this research field. Among others, on-board rotors can undergo parametric instabilities as soon as they are subject to a rotational motion of their base. Nevertheless, all the cases of study available in the literature are only limited to mono-axis rotation of the base or at least multi-axes rotations with the same parametric frequency. In practice, the excitations are not generally restricted to such cases. In this context, the present paper aims to investigate the dynamics of an on-board rotor subject to multi-axes sinusoidal rotations of its base, with arbitrary frequencies. This is achieved by analyzing the finite element model of an on-board rotor, which is composed of a slender shaft with two disks, supported by two hydrodynamic bearings having finite length. The nonlinear forces related to these bearings are linearized in the vicinity of the rotor static equilibrium position for a constant speed of rotation so as to obtain a linear system. Floquet theory is finally applied in order to perform stability analysis and the influence of the amplitudes, frequencies and phases of the excitation is assessed. This rotordynamics prediction is essential for the design of many on-board rotating machineries such as those implemented in vessels subject to roll or pitch motions induced by the waves to name just one example.