Wikpedia defines "parametric statistics" as:
...a branch of statistics which assumes that sample data comes from a population that follows a probability distribution based on a fixed set of parameters.
Many hypothesis tests usually characterized as parametric, such as the classical one-sample $t$-test for the sample mean, are based on knowledge of a parametric distribution for the statistic. In terms of assumptions on the data, the $t$-test requires that the data be normally distributed only if the sample is small, but once the CLT kicks in, no distributional assumptions on the data are required.
Would it therefore be fair to describe the $t$-test and similar as "parametric in finite samples but asymptotically nonparametric"? This would seem to be a very unusual characterization.