In engineering systems, many parametrically excited vibrations can be modeled by the Mathieu equation with nonlinear terms. This study establishes a physical model using a bass drum trigger, where the striking process of the drum beater is equivalently represented as a Kapitza-like pendulum system. Through Taylor expansion, we derive a Mathieu-Duffing equation incorporating a quintic nonlinear term. A perturbation method is systematically employed to analyze the transition curves and stability variations near the first parametric resonance tongue, revealing the coexistence of multiple equilibrium points in the system. By investigating the eigenvalues of the Jacobian matrix, we explicitly demonstrate the emergence mechanisms of both supercritical and subcritical pitchfork bifurcations at equilibrium points, with particular emphasis on their intrinsic relationships with nonlinear parameters. Numerical simulations are conducted to reconstruct the bifurcation diagram within the three-dimensional parameter state space for the first resonance tongue, thereby validating the theoretical predictions. The results comprehensively illustrate the intricate interplay between parametric excitation, nonlinear stiffness, and bifurcation dynamics in such hybrid oscillator systems.