A general linear model is a generalized linear model but not vice versa. A generalized linear model can use any number of sampling distributions where as the general linear model works of the Gaussian distribution.

This is not a correct assumption. Linear models assume the population distribution you draw from is normal not the data itself. To check this you must first run the model and then look at a QQ-plot of the error terms (residuals) to detect non-normality, then you make decisions about parametric vs. non-parametric.I have some data that I want to analyse (Q-Q plots show no normal distribution of data), which I assumed have to be analyzed by non-parametric methods.

If your data does come from a normal population distribution then the parametric test will most likely give you the greatest power, not the no-nparametric tests as they are more flexible and make less assumptions but also give you less power to reject the null hypothesis.