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?

There are numerous examples of parametric tests that have nothing to do with normality.  In fact, the underpinning of the hypothesis testing theory: Neyman-Pearson lemma applies to testing any parametric probability density $\rho$. The backbone of the hypothesis testing theory is likelihood ratio, not normality.
On the other hand, you got this impression probably because with the normality assumption, the sampling distribution of the test statistic (under the null hypothesis) becomes mathematically tractable.  This is why most of the classical hypothesis testings that you learned from class relied on the normality assumption.  However, as stated above, this by no means rules out the possibility of doing hypothesis testing with other parametric distribution families.
