The equation of state of matter at supranuclear densities governs the astrophysical observables of neutron stars. A realistic, though complex, description is provided by the Chiral-Mean-Field model, which depends on many microscopic nuclear-physics parameters. We present a Fisher-information-inspired analysis of the sensitivity of neutron-star observables to the parameters of the Chiral-Mean-Field model for cold, charge-neutral, $\beta$-equilibrated neutron-proton-electron matter connected to a SLy crust. We then compute neutron-star sequences and extract masses, radii, compactnesses, and tidal deformabilities. From the logarithmic derivatives of these observables with respect to each nuclear parameter, we construct a dimensionless, Fisher-inspired sensitivity matrix and perform a principal-component analysis to identify the effective combinations of nuclear parameters that most strongly affect neutron-star observables. Although the ranking depends mildly on the observable, the three most important nuclear parameters are the vacuum value of the dilaton field ${\chi }_0$ (which sets the overall scale of the scalar potential and trace-anomaly contribution), the scalar singlet strength $g_1^X$ (which controls the overall scalar attraction through the baryon effective masses), and the $k_0$ quadratic scalar term (which governs the curvature of the scalar potential). This dimensional reduction should be understood as a local active-subspace statement around the fiducial CMF C4 parametrization, rather than as a global restriction on the EoS flexibility of the CMF framework. Nevertheless, it provides a reproducible, data-driven approach to quantify parameter sensitivities in dense-matter models and to guide future Bayesian inference of nuclear information from multi-messenger astrophysical observations.