Gaussian distribution is a common choice when dealing with symmetric data. However, other alternatives must be considered in applications with high tail-weight. One option is the randomization of the scale parameter for the Gaussian distribution, enabling a more flexible model for the tails albeit maintaining symmetry. Although any positive random variable can be used as a random scale parameter, Pareto distribution is a suitable choice in order to increase variance and tail-weight. Therefore, the aim of this work is to study the Pareto randomization of the scale parameter for symmetric distributions, in particular for the Gaussian distribution. Estimation problem is tackled and a simulation study is discussed. Finally, an application concerning the directions chosen by ants after a stimulus is provided. The results reveal that the proposed methodology works well both on simulated and real data.