In this paper, the reliability of a linear oscillator subjected to non-stationary seismic excitation is estimated considering the uncertainties of the parameters in the structure and input model. The input is described by a seismological model with evolutionary power spectral density. The probability distributions of the parameters in the input model are estimated from 54 actual accelerograms through least-square linear and non-linear regression analyses. The statistics of the input model parameters thus obtained and the statistics of the structural parameters, adopted from available data in the literature, are considered in estimating the time-variant reliablity of the oscillator through Yang's Markovian extreme point process. Two different methods, namely, the method of moments and the Advanced First-Order Second-Moment method, are used. The results from an example show that the variabilities of the structural and input model parameters can contribute up to 96% of the variance of the maximum peak of the response, and that changes in the seismological parameters affect the oscillator's reliability more than changes in the structural parameters. This observation, together with the results from sensitivity analysis on the parameters, indicate the need for better estimation of the input model parameters.