Difference between parametric and nonparametric statistical tests and What is the difference between distribution free statistics/methods and non-parametric statistics? is a good read for beginners. So, does system identification of linear models like Auto regressive, Moving AVerage, ARMA And learning of neural networks by estimating its weights fall under Parametric or non parametric methods?
Artificial neural networks are nonparametric.
AR/MA/ARMA models may most often be estimated with some parametric assumptions, but nonparametric methods are available for their estimation as well (e.g., Holzberger's Nonparametric Estimation of Nonlinear ARMA and GARCH Processes). Thus for ARMA models, either may be used.