The resonance response of a single-degree-of-freedom nonlinear dry oscillator of Coulomb type to narrow-band random parameter excitation is investigated. The analysis is based on the Krylov-Bogoliubov averaging method. The averaged equations are solved exactly and the algebraic equation of the amplitude of the response is obtained in the case without random disorder. Linearization method and moment method are used to obtain the mean square response amplitude for the case with random disorder. The effects of damping, nonlinear intensity, detuning, bandwidth, dry intensity, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced and the bifurcation of the system will be delayed when intensity of the nonlinearity increases. The peak amplitudes will also be reduced and the bifurcation of the system will be delayed when damping and dry intensity of the system increases.