This is why I think that modeling elementary physics is nothing else than fitting data. We might end up with something that we perceive as "simple", or not. But in any case all the fitting has been hidden in the process of ruling out models. It's just that a lot of the fitting process is (implicitly) being done by theorists; we come up with new models and that are then being falsified.
For example, how many parameters does the Standard Model have? It's not clear what you count as a parameter. Do you count the group structure, the other mathematical structure that has been "fitted" through decades of comparisons with experiments?
You are using the word "fitting" rather loosely. We usually "fit" models of fixed function form and fixed number of parameters.
You are also glossing over centuries of precedent that predate high-energy physics, namely quantum field theory, special relativity, and foundational principles such as conservation of energy and momentum.
It tends to be a parameter that can be derived from rrasoning and assumptions. This contrasts to free parameters where you say "and we have no idea what this value should be, so we'll measure it"
