This study proposes a model-based clustering framework for time series by incorporating Laplace and Skew-Laplace distributions to address non-normality, skewness, and heavy-tailedness in empirical data. The proposed approach assumes that cluster observations follow a polynomial regression characterized by a temporal Vandermonde design matrix. Model parameters are estimated using both Frequentist and Bayesian frameworks. Simulation results demonstrate that while the Skew-Laplace model performs comparably to Normal and Laplace models under symmetric conditions, it significantly outperforms them when the data exhibit skewness and heavy tails. In such scenarios, symmetric distributions typically overestimate variance to compensate for their lack of flexibility, whereas the Skew-Laplace model provides a robust and efficient fit. Furthermore, the Bayesian approach exhibits superior performance over frequentist methods, particularly in clusters with limited sample sizes. These findings underscore the effectiveness of the Skew-Laplace model as a comprehensive framework for complex time series analysis, offering high adaptability across diverse data structures.