Does one need to specify a probability distribution in parametric time series model? It seems to me that one doesn’t need to specify a probability distribution in time series model but only something like AR1 or ARMA. Is there some kind of non parametric time series model where even the AR1 or MA specification isn’t needed? 
 A: It depends on your need. 
In fact, the AR(p) model can be estimated using OLS and without assuming any distribution. 
However, more complex models such as ARMA you need a specific distribution for the past inovations (errors). So, the assumption of the distribution is necessary because you need some extra information to estimate the parameters.
It is the same with  simple linear regression model. You do not need a distribution to estimate it. As long as your error is spherical, you can use the OLS to estimate (if your error is not spherical, for instance, different observations have different variances, you need something that account for this fact). However, if you need to do some inferencial statistics such as to check if the null hypothesis is valid or not, you need to assume a distribution for the error.
In general, if you are dealing with forecast, you usualy do not need a distribution associated with your model, since you can check whether your model 
is good or not using cross validation. This is the largest difference between machine learning and statistical inference results.
