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.

You apply parametric tests under the assumption that the parametric model is right. This always greatly constrains the set of possibilities you are considering. Hence the power.
Consider a parametric bootstrapping where you constrain all possible distributions to a particular set of distributions such as normal. So instead of infinite set of all possible distributions you are only looking at Gaussian with just two parameters. Naturally, the tests you can come up with are going to be sharper.
Remember that the power comes from the assumptions, and unfortunately, assumptions are often wrong. If your assumption is wrong then the power evaporates. No free lunch here
