Our course slides offer the following definition: "A parametric probabilistic model is a set of probability distributions indexed by a finite-dimensional parameter vector."
This description defines "parametric probabilistic model" in terms of "parameter", which isn't particularly helpful. But it brings to mind some vector of indices which tells you where to find a row (or rows) in a table.
It is not clear to me whether "parameter" in this sense is related to my CS understanding of a parameter (i.e., a "knob") or even if a model's parameters have something to do with its random variables (e.g., are the parameters fixed values of a random variable?). Furthermore, what does it mean to "parameterize" a model?