In power transmission systems, conductor vibration may cause periodic variations in the axial load carried by composite suspension insulator strings. Such axial-load variations may induce parametric vibration, affecting the mechanical reliability of transmission lines and the safe operation of transmission networks. This study examines how excitation frequency and amplitude affect the parametric response of a reference core-rod model for an FXBZ-500/300 composite suspension insulator. The core rod is modeled as a uniform Euler–Bernoulli beam, and the resulting eigenvalue problem is solved numerically. Its lateral restraint is calculated from published conductor data and stated support assumptions. A first-mode Galerkin reduction shows that the fluctuating axial load modulates the system stiffness through a time-varying geometric-stiffness term. The resulting undamped equation takes the Mathieu form, with the primary instability region centered near $f_{\Omega }=2f_1$. At exact resonance, the linearized damped model predicts a damping-dependent load-amplitude threshold. The first six natural frequencies predicted by the analytical beam model differ by less than $0.7\%$ from those obtained from the prestressed finite-element model, indicating numerical consistency between the two idealized beam models. For the prescribed small initial disturbance, the tested cases show decay below this threshold and bounded finite-amplitude first-mode oscillations above it. As the dynamic-load amplitude increases, the calculated midpoint-displacement amplitude increases and the response peak shifts to a higher excitation frequency, consistent with stretching-induced hardening under the fixed-end-separation reference assumption. Within the adopted model, these results identify combinations of axial-load frequency and amplitude that produce pronounced core-rod vibration. Their relevance to an operating line depends on a sustained axial-load component near $2f_1$ being transmitted to the string.