The Pearson correlation coefficient is sometimes referred to as a parametric statistic. Does this parametric nature imply that it is actually only applicable to data drawn from Gaussian distributions?
Correlation is a simple metric, such as variance and mean. These metrics can be applied to any type of data, normal or not. Pearson's correlation can be described as either a Method of Moments estimator (in which case you would say there's no connection to a bivariate Gaussian) or as a maximum likelihood estimator for the correlation parameter for a bivariate Gaussian (which does imply bivariate Gaussian).
The implications of this is that if your data really comes from a bivariate Gaussian, then Pearson's correlation is an asymptotically optimal way to look for the correlation between two variables (i.e., it is the MLE). If your data is not bivariate Gaussian, you are still using a method that consistently estimates the correlation between two variables, but it's possible that these two variables may have non-linear dependencies that Pearson's correlation may have a hard time picking up.
If you believe that there is a monotonic but non-linear relation between two variables, then Spearman's Rho is commonly used. It's my somewhat limited understanding that distance correlations can be used to look for non-monotonic dependencies, but to be honest I'm not super familiar with these methods at this time.