Are there any parametric machine learnig models that can represent non-linear behaviour between a predictor and a targe variable? Some basic models, like multiple linear regression (MLR), are parametric but usually limited to linear interactions between X and y. Other models can represent non-linear interactions between X and y, but to my understanding, most of these are non-parametric, such as Random Forest. I wonder if there are parametric models capable of embedding non-linear behaviour. I thought of some model that would assume polynomials in its structure, but I did not find anything.

You can transform your original predictors in different ways, e.g., using splines or sine/cosine transformations, then feed them into regression-type models (OLS, GAMs, ...). Frank Harrell's Regression Modeling Strategies discusses this at length.
