A parametric test is making an assumption about the distribution of the random variable, not it's mean.
Inside many of these tests, they might make use of CLT assumptions of sample means, but that doesn't mean they don't still depend on the distribution assumptions.
Skewness is a major consideration and can lead to completely different inferences.
Let's say we wanted to find the median and our distribution was assumed to be normal. Under no skewness, the sample mean would be a good approximation. Under skewness the sample mean would be a very bad approximation.
If the test in question is solely about the mean of the random variable, and nothing else about the distribution, then it's possible that the normality assumption only need to extend as far as the sample mean (ala t-test). But that's hardly a parametric test anymore is it?
It's not clear to me what kinds of tests you are referring to. Ordinary least squares regression, for example, is all about estimating conditional means, and it is very much parametric. Just finding the best estimates for the parameters of a one-dimensional distribution is usually not particularly interesting, is certainly not what statisticians spend most of their time on and in any case nobody's suggesting that the population mean is always equal to the population median.
Why do you think we have the classification 'parametric' if the only thing that matters is the distribution of the sample mean? If it's all going to converge to be normal as you say, why is there parametric and non-parametric tests?
Regression works like this: E[Y|X] = Xβ. It is parametric because you model the conditional mean as a weighted sum of various predictors, and these beta "weights" are your parameters. This is true of ordinary regression, Poisson regression, binomial (logistic) regression and so on. An example of nonparametric regression would be something like regression splines.
Why are there nonparametric tests? Because for small sample sizes you can't always trust the normal approximation, and as you state this might be due to something like skew. This takes nothing away from the fact that inferential statistics is almost always about comparisons of means. And yes, the t-test is a parametric test, of which the Mann-Whitney or Wilcoxon would be the nonparametric equivalents.
Inside many of these tests, they might make use of CLT assumptions of sample means, but that doesn't mean they don't still depend on the distribution assumptions.
Skewness is a major consideration and can lead to completely different inferences.
Let's say we wanted to find the median and our distribution was assumed to be normal. Under no skewness, the sample mean would be a good approximation. Under skewness the sample mean would be a very bad approximation.
If the test in question is solely about the mean of the random variable, and nothing else about the distribution, then it's possible that the normality assumption only need to extend as far as the sample mean (ala t-test). But that's hardly a parametric test anymore is it?