This paper proposes an estimation approach of the Whittle estimator to fit periodic autoregressive moving average (PARMA) models when the process is contaminated with additive outliers and/or has heavy-tailed noise. It is derived by replacing the ordinary Fourier transform with the non-linear M-regression estimator in the harmonic regression equation that leads to the classical periodogram. A Monte Carlo experiment is conducted to study the finite sample size of the proposed estimator under the scenarios of contaminated and non-contaminated series. The proposed estimation method is applied to fit a PARMA model to the sulfur dioxide (SO2) daily average pollutant concentrations in the city of Vitória (ES), Brazil.