Abstract <p>The parametric roll instability observed in wave-tank experiments of a point-absorber wave energy converter (PA-WEC) is investigated using a nonlinear multi-degree-of-freedom (multi-DoF) mathematical model based on the Lagrange equation of the second kind. The instability is caused by the oscillatory variation of the pendulum length. The model accurately reproduces the roll instability observed in experiments and reveals that the roll motion eventually converges to a stable limit cycle. The nonlinear model is reduced to a third-order augmented Mathieu equation without weakening its essential features. Nonlinear asymptotic analyses are conducted on the third-order augmented Mathieu equation to determine the stable and unstable regions in its parameter space. The effect of the damping term on the solution behavior is also presented. The analytical solutions provide insights into the contributions of different terms across multiple time scales. The multi-scale expansion method is further employed to derive the explicit analytical solutions of the third-order augmented Mathieu equation, and to provide analytical insights across different time scales. These findings not only deepen the understanding of the nonlinear dynamics of PA-WECs but also inform practical applications, such as optimizing energy harvesting and designing control strategies to enhance survivability and performance under parametric resonance conditions.</p> Graphical abstract <p></p>