<p>This paper investigates the stability, parametric resonances, and complex dynamics of a Duffing-type oscillator with periodic modulation acting through the inertial term. In contrast to the conventional Mathieu-type setting, the present model exhibits a distinct resonance structure because the parametric excitation enters the coefficient of the highest-order derivative rather than the stiffness term. To analyze the weakly nonlinear dynamics, the method of multiple scales is employed to derive solvability conditions and Landau-type amplitude equations for the non-resonant, super-harmonic, sub-harmonic, and primary parametric-resonance cases. These reduced equations are then used to obtain analytical approximations of the transition curves and stability boundaries, thereby clarifying the effects of damping and cubic nonlinearity on the onset and deformation of resonance regions. To complement the local asymptotic analysis, the global nonlinear response of the forced system is examined numerically through bifurcation diagrams, largest Lyapunov exponents, phase portraits, time histories, and Poincaré sections. The results reveal successive period-doubling bifurcations leading to chaos and demonstrate how parametric modulation and Duffing nonlinearity reshape both the stability boundaries and the strongly nonlinear response patterns. Overall, the study provides an integrated analytical and numerical investigation of the oscillator dynamics and highlights the interplay between local resonance mechanisms and global nonlinear behavior.</p>