Data is neither parameteric or nonparametric. Data is data. An analysis method might make some assumptions about distributions involving parameters in which case we call the method parametric. There are methods that don't make any distributional assumptions (or at least very few assumptions about the distribution) and we call those nonparametric.
Now when doing a regression analysis we typically assume that the error term is normally distributed. There isn't an assumption on the distribution of the predictor variables and note that we don't actually care about the marginal distribution of the response either. The way to check this assumption is to look at the residuals from the regression and check if those look approximately normally distributed (because the residuals are sort of what we're using to represent the error term). However, if all you're looking to do is get the OLS fit then you don't actually need to make the assumption of normally distributed errors - we typically just make that assumption so we can do inference on the parameters.
There are other types of regressions where you don't need to assume the errors are normally distributed.