Why are parametric tests more powerful than non-parametric tests? I'd like to understand why parametric tests are more powerful than their non-parametric alternatives. Is the word choice of "power" the same as statistical power? As I understand it, power just relates to the likelihood of getting a p-value that will correctly reject a false/incorrect null hypothesis, but I don't understand how this relates to statistical tests based on the normal distribution specifically.

Parametric methods can be more powerful than non-parametric in some circumstances, but are not universally so. Even when the circumstances most strongly favour the parametric approach the power advantage is often minor or even trivial.
When parametric methods have an advantage in power it comes from one or both of two things: more information in the statistical model (e.g. knowing the distribution type of the population can allow the sample estimates to be more efficiently used); more information extracted from the data (e.g. when it is appropriate, using the actual numbers is more informative than using ranks).
It is important to know that when the statistical model assumptions are invalid then the parametric methods may have less power than a well-chosen non-parametric approach.
