Do parametric tests make assumptions regarding linearity? Whether yes or no, how can I explain this? It seems to me as though regression analyses assume linearity, but I can't seem to figure out whether these can be parametric tests or not.

Many thanks!

Yes, indeed. And, the Analysis of Covariance (ANCOVA) would be a good example; which is a combination of ANOVA and Regression. The relationship between the variate (dependent variable Y) and the covariate (X) is traditionally assumed to be linear across the treatment groups.

That said, the relationship between Y and X in ANCOVA could be non-linear e.g. quadratic - where the square of the covariate (X) would become the second covariate in the ANCOVA model. More generally, this is also used in curvilinear (or polynomial) regression analyses, where F-statistics can be computed in a hierarchical manner to determine the "deviation from linearity" when X is quantitative (not qualitative) in nature e.g. the amount (mg) of a drug dosage, the amount of hours of sleep deprivation, the number of hours of practice time, and so on.