Most of the publications I have reviewed, conduct both OLS and GWR for comparison, and prior to analysis, ensure dependent and explanatory variables are normally distributed. Can someone explain why this is necessary for GWR? Are both OLS and GWR parametric? Is there a non-parametric version of GWR?
No, GWR is a frequentist, linear method but there is flexibility in how you define the distributional form of the model (ie., Poisson, Binomial, Gaussian). The GWR is a local regression that emphases 2nd order variation whereas OLS is a first order model. The general motivation in running both is to draw inference about first (global) and second (local) order process but, more directly GWR is specified to account for nonstationarity.
As far a normality, heteroscedasticity is impossible to evaluate in a local regression and as such is effectively ignored. However, this is important in the OLS.
Please note that GWR is a very suspect method and unless you have very strong nonstationarity in your data I would highly recommend abandoning its application. Besides, in the absence of nonstationarity, there is really no justification for the specification of a local regression. In the absence of second-order variation GWR will likely just overfit the data.
An OLS can often be robust to weak first-order spatial autocorrelation and should provide a reasonable model. If not your next step should not be GWR but a model, such as mixed effects models or spatial autoregressive model, where a spatial term can be explicitly defined.