As offshore oil exploration goes into deeper waters, spar platforms appear as a good alternative for oil field developments due to their inherent hydrodynamic behaviour regarding its vertical motions response in waves. Spar’s long natural periods in heave and pitch guarantee the good linear responses of the associated motions; however, nonlinear unstable motions can be triggered. Many numerical and experimental investigations have put forward the reasoning that this kind of floating structure is prone to parametrically induced motions due to changes in their pure hydrostatic restoring. Based on an analytical model, the present paper demonstrates that the main contribution to parametric excitation comes from variations in the pressure field associated to the incident wave and not from purely nonlinear hydrostatic actions. A nonlinear mathematical model based on Taylor series expansions, Neves and Rodríguez (Ocean Eng 33(14):1853–1883, 2006), is employed to explain the underlying mechanism that leads to the phenomenon of Mathieu instability in vertical deep drafted cylinders. Analytical expressions are derived for the nonlinear hydrostatic and Froude-Krilov actions. A set of coupled time-dependent equations is obtained. Based on that, general conditions for the appearance of principal resonances are derived. These analytical results are verified by means of numerical simulations. Numerical analyses are carried out for regular wave conditions and a domain of parametric amplifications is obtained.