Is a p-value a sample statistic, or a population parameter, or neither? I would also be interested to know why it is one or the other.
 A: A $p$-value is the probability of observing a test static value as or more extreme than the test statistic created from one's data if the null hypothesis is true. You can therefore interpret the $p$-value as a measure of how extreme your test statistic is under H$_{0}$ and the probability distribution attached to H$_{0}$. The $p$-value is therefore a statistic that is a function of one's data, and one's choice of H$_{0}$.
A: If the test statistic can be called a statistic, then so must the $p$-value:  the test statistic is a function of the data under the assumption that the null hypothesis is true.  The $p$-value is simply a probability associated with that test statistic.
A: A p-value is a random variable, so it's not a population parameter.
You could certainly argue that it's a statistic:

A statistic (singular) is a single measure of some attribute of a sample (e.g., its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items of the sample, which are known together as a set of data.

