The following issue with the use of parameters in model theory has been one of the greatest challenges in my attempt to learn model theory.
Based on what I have seen, I understand that model theorists often do not make explicit the set to which the parameters of formulas and types belong when they know what they are talking about. I have also observed that they do this when they teach model theory.
For instance, consider the following definition of forking the likes of which I have seen in multiple textbooks of model theory:
A formula $\phi(x, a)$ forks over $B$ if $\phi(x, a)$ implies a finite disjunction $\bigvee_i \psi_i(x, b_i)$ of formulas $\psi_i(x, b_i)$ that divide over $B$.
I asked myself, where do the parameters of $\psi_i$ belong? Are they formulas over $B$?
Of course, a model theorist would immediately say that, since no formula over $B$ divides over $B$, the formulas $\psi_i$ are meant to have parameters from elsewhere (anywhere in the monster model?).
I don't know model theory, however, and that is the very reason I read those textbooks.
The question I would like to ask is how I can correctly guess the set from which model theorists take parameters.
Addendum: This is not a question how parameters work in general! This is not a question on definitions of forking either, although I would appreciate comments that clarifies the definition above.