This is a good question. Frequently, one will see smoothing regressions (e.g., splines, but also smoothing GAMs, running lines, LOWESS, etc.) described a **nonparametric regression** models.

These models are nonparametric in the sense that using them does not involve reported quantities like $\widehat{\beta}, \widehat{\theta}$, etc. (in contrast to linear regression, GLM, etc.). These models are extremely flexible ways to represent properties of $y$ conditional on one or more $x$ variables, and do not make *a priori* commitments to, for example, linearity, simply integer polynomial, or similar functional forms relating $y$ to $x$.

On the other hand, these models **are** parametric, in the mathematical sense that they indeed involve parameters: number of splines, functional form of splines, arrangement of splines, weighting function for data fed to splines, etc. In application, however, these parameters are generally not of substantive interest: they are not the exciting bit of evidence reported by researchers… the smoothed curves (along with CIs and measures of model fit based on deviation of observed values from the curves) *are* the evidentiary bits. One motivation for this agnosticism about the actual parameters underlying a smoothing model is that different smoothing algorithms tend to give pretty similar results (see Buja, A., Hastie, T., & Tibshirani, R. (1989). [Linear Smoothers and Additive Models](https://projecteuclid.org/download/pdf_1/euclid.aos/1176347115). *The Annals of Statistics*, 17(2), 453–510 for a good comparison of several).

If I understand you, your "mixed" approaches are what are called "semi-parametric models". [Cox regression](https://en.wikipedia.org/wiki/Proportional_hazards_model) is one highly-specialized example of such: the hazard function relies on a nonparametric estimator, while the explanatory variables are estimated in a parametric fashion. GAMs—generalized additive models—permit us to decide which $x$ variables' effects on $y$ we will model using smoothers, which we will model using parametric specifications, and which we will model using both all in a single regression.