We propose a new periodic autoregressive model for seasonally observed time series, where the number of seasons can potentially be very large. The main novelty is that we collect the periodic coefficients in a second-level stochastic model. This leads to a random-coefficient periodic autoregression with a substantial reduction in the number of parameters to be estimated. We discuss representation, parameter estimation, and inference. An illustration for monthly growth rates of US industrial production shows the merits of the new model specification.