This paper establishes the statistical properties of a spectrum-based Whittle parameter estimation procedure for long-range dependent locally stationary processes. Both theoretical and empirical behaviors are investigated. In particular, a central limit theorem for the Whittle likelihood estimation method is derived under mild distributional conditions, extending its application to a wide range of non-Gaussian time series. The finite sample properties of the estimators are examined via Monte Carlo experiments with Gamma and Log-normal noise distributions. These simulation studies demonstrate that the proposed method behaves properly even for small to moderate sample sizes. Finally, the practical application of this methodology is illustrated by means a well-known real-life data example.