In 2020, the IMO issued Interim Guidelines on the Second-Generation Intact Stability Criteria for Ships. To support the implementation of these criteria in the shipbuilding industry, ship design software requires time domain numerical methods of parametric roll with varying levels of accuracy and computational efficiency. First, a time-domain parametric roll mathematical model coupling heave, roll, and pitch in three degrees of freedom is developed based on two-dimensional and three-dimensional hydrodynamic methods. Second, in the two-dimensional hydrodynamic method, time-domain heave and pitch motions are obtained using a frequency-to-time domain conversion method, while in the three-dimensional hydrodynamic method, the three-dimensional time-domain mixed-source method is employed to directly solve for heave and pitch motions in the time domain. Subsequently, the Froude-Krylov moments and hydrostatic moments in the roll direction are solved along the instantaneous wet surface. Finally, using the international standard C11 container ship model, the effects of heave and pitch motions obtained from frequency-domain solutions, time-domain average wet surface solutions, and time-domain instantaneous wet surface solutions on parametric roll are compared, and the results are compared with model test data. It is verified that the three-degree-of-freedom mathematical models based on two- and three-dimensional hydrodynamic methods can quantitatively predict the time-domain motion of parametric roll in head waves.