Example of a parametric test with no normality assumptions? I read an article saying that "all parametric tests make an assumption of normality", which I know to be incorrect, since technically a parametric test could be done for any distribution, but I couldn't actually find any tests that didn't make an assumption of normality somewhere in them.
Are there any examples of a parametric test that makes no assumptions about normality at all?

Just to give a concrete example of a parametric test not assuming normality: the binomial test as implemented in binom.test function in R. It's just a test for one proportion that uses the binomial distribution instead of approximating it by a normal distribution (thus requiring normality) as it's usually done in elementary statistics courses.
You can see that the p-value given by binom.test is exactly the same we get using the binomial distribution:
> binom.test(5,20,.3, alternative = "less")

    Exact binomial test

data:  5 and 20
number of successes = 5, number of trials = 20, p-value = 0.4164
alternative hypothesis: true probability of success is less than 0.3
95 percent confidence interval:
 0.0000000 0.4555824
sample estimates:
probability of success 
                  0.25 

> pbinom(5, 20, .3)
[1] 0.4163708 

The Poisson test (implemented in poisson.test in R) would be a similar example of parametric test that does not assume normality.
