I am a little bit familiar with quantile regression but I was wondering what is the difference between parametric quantile vs non-parametric quantile. Also, the "rq" function is R is based on parametric version or nonparametric version?
I think in the context of Koenker's quantreg package, he meant non parametric for non parametric regression, which means the form of covariates is not fixed. Rq assumes a linear form so it's not non parametric. On the other hand, rqss uses a penalized approach so is non parametric in that sense.
To note, you can also say non parametric when referring to a distribution of random variable as was pointed out in comments but here, due to regression framework, there are parameters to estimate making this a semi parametric, not non parametric.
To add, I've also came across some literature assuming some distribution e.g. Laplace, normal mixtures to force a likelihood inference. In that sense, these are fully parametric models like classical mean regression.