I am dealing with non-normally distributed data, which one would be better to perform non-parametric or the parametric analysis?
There is no particular answer to this question, using parametric or non-parametric analysis depends on the questions and details of your research design. Non-parametric tests are not assumption free. Their assumptions are dependent on the context. On the other hand, Non-parametric tests have a broad class of models available. It deals with different types and distributions.
The decision of using parametric or non-parametric is dependent on the fundamental change in the test power. If the study is not getting enough transformation, then, non-parametric would be used, Otherwise parametric tests. Non-parametric tests strongly violate the assumption of normality.