Parametric does NOT mean "Bayesian based".
Here is one definition of "parametric statistics"
Parametric statistics is a branch of statistics that assumes data come
from a type of probability distribution and makes inferences about the parameters of the distribution
As Wikipedia goes on to note, most of the common, elementary statistics are parametric. For example, ordinary least squares regression is parametric. Loess regression is nonparametric.
Parametric statistics are usually easier to interpret and may be more powerful (in a statistical sense) but they are based on more assumptions than nonparametric statistics. They vary in their degree of robustness, but are usually less robust than nonparametric statistics.
For example, the equation derived from ordinary least squares regression is (in most cases, anyway) quite easy to understand. That from a regression involving splines is often much less clear and may require graphical representation to be understood well.
Bayesian statistics is something altogether different, having to do with using prior information.