"Highly parameterized" meaning O(20) free parameters. It matches thousands upon thousands of detailed precision data points.
But yeah, I agree that the "highly parameterized" part is a statement from fashion, and the number of parameters is really not a good reason to try to replace the Standard Model. (There are many good reasons, but this one isn't one of them.)
Also, I am yet to see any alternative proposal with fewer parameters.
Fine tuning, I agree, is a philosophical issue. I'm a physicist, and I don't buy it. Why does everything have to be perturbatively pleasant? Nobody promised us that.
The issue of artificial parameters is a red herring, I think. Properly computed, of course, well-defined observables are renormalization scale independent. You might have to pick a scheme/scale to do the calculation, but whatever scale dependence remains is an indication of some perturbative truncation. The continuum limit of LQCD, for example, produces real observables with no renormalization scale dependence. Hell, renormalization is not even mysterious in a computational approach.