I echo much of your concerns and observations in the comments!
Parameter estimator is a tool to find a minimum by tuning the parameters we tell it to tune, to match a set of data we tell it to match, starting from initial values we tell it to start from, and bounded by a set of min/max that we tell it as the boundary. It's not a "smart" tool in such sense; it's just an automated tool. Lots of insights from the users, as domain experts of the system we are modeling, are needed to make the applicaiton of this tool successful.
There is a sensitivity analysis tool within the Estimator that can help you identify the sensitive parameters, BUT it probably can't distinguish between "parameters that depend from combinations of parmeters". Judgment from domain experts cannot be replaced here. I usually rely on my understanding of the physical system and my understanding on the effect of certain parameters, when choosing the parameters to tune and to verify that the provided reference data is sufficient to tune those parameters.
The outcome, as a combination of estimated values parameters, as you hinted in your coments, is very likely not unique, which means they may result in match with the reference data but they may not represent the real values of the parameters. One way to avoid this is to always have a set of test data, independent of the experiment data used to tune the parameters, to test whether the set of parameters works in other cases. Parameter Estimator support testing with test data and I would recommend using that whenever possible.
My thoughts so far. Proceed with caution :)
