I don't follow you. Overfitting happens when your model has too many parameters, relative to the amount of data you have. It is true that linear models may have few parameters compared to some non-linear models (for example linear regression models vs regression models with extra interaction parameters). But surely, we can have sparsely parameterized non-linear models as well.

Sure, technically if Alice fits a small noisy data set as y(x) = a*x+b and Bob fits it as y(x) = c*Ai(d*x) (where Ai is the Airy function) they've used the same number of parameters, but that won't stop me from rolling my eyes at the latter unless he has a good first-principle reason to privilege the hypothesis.