I am looking for parametric functions with parameters to control its nonlinearity?
For a trivial example, $$f(x) = (1-\alpha)x + \alpha g(x)$$ where $0\leq\alpha\leq 1$ and $g(x)$ is an arbitrary nonlinear function.
Have you observed some more interesting functions with this property?
Let $f(x,\alpha)$ be 0 if $x$ is rational, and $\alpha$ otherwise.
But honestly, if you don't define something which must be fulfilled to e.g. conclude that $f(x,0.1)$ is "more linear" than $f(x,0.2)$, the whole thing is a bit arbitrary (note that in the common understanding, a function is either linear or not, and that there is nothing in between).