Serial correlation and overdispersion must be handled properly in analyses of time series of counts, and parameter-driven models combine an underlying latent process with a conditional log-linear Poisson model (given the latent process) for that purpose. Regression coefficients have direct interpretations, but likelihood inference is not straight-forward. We consider a two-step procedure for estimation: First regression parameters are estimated from the marginal distribution; second parameters concerning the latent process are estimated with composite likelihood methods, based on low-order simultaneous or conditional distributions. Confidence intervals are computed by bootstrap. Properties of estimators are examined and compared to other methods in three simulation studies, and the methods are applied to two datasets from the literature concerning hospital admission related to asthma and traffic deaths.