A different and somewhat more abstract viewpoint is given by considering the "squared distance function" defined by $\rho(x,y)=d((x,y),G)^2$ for all $(x,y)\in \Bbb R^2$. Here $G=\{(u,v):v=f(u)\}$ is the graph of your function, considered as a subset of $\Bbb R^2$, and $d((x,y),G)$ is the distance from $(x,y)$ to $G$, which is the same as the distance from $(x,y)$ to the closest point in $G$. (Assume that your function $f$ is continuous on $\Bbb R$ so that $G$ is a closed subset of $\Bbb R^2$.)
If your function $f$ is smooth then $\rho(x,y)$ will be smooth on a neighborhood of $G$. More precisely, if $f$ is smooth of class $C^k$ with $k\geq 2$ then $\rho(x,y)$ will be smooth of class $C^k$ near the graph $G$. (See this.) From now on, we always assume $k\geq 2$. We have used the squared distance function to get smoothness on the graph $G$, in the same way that the function $x^2$ is smooth at $x=0$ wheras the function $|x|$ is not.
How far away from $G$ will $\rho(x,y)$ be smooth? Let $(x,y)$ be some point in $\Bbb R^2$ and let $(u,v)$ be the point on $G$ which is closest to $(x,y)$ (assume that there is only one such point). If the distance between the two points is less than the radius of curvature of $G$ at $(u,v)$ then we are guaranteed that the squared distance function will be smooth at $(x,y)$.
Now compute the radius of curvature $r(u,v)$ of $G$ at a general point $(u,v)$ on the graph $G$. If the radius of curvature is bounded from below on $G$, so that we have $r(u,v)\geq c$ for some $c>0$ and all $(u,v)$ on the graph, then the squared distance will be smooth on the set of points $\{(x,y):d((x,y),G). You can then define parallel graphs on this set as level curves for the distance function $d(\cdot,G)$.
Can something go wrong here? Yes, if there are more than one point on $G$ which is nearest to $(x,y)$. For general curves this can be a problem, but since your curve is the graph of a function this problem cannot occur when $d(x,y), where $c$ is the uniform lower bound from the last paragraph for the radius of curvature.
I called this approach more abstract, since it is not so easy to get explicit formulas for the distance function $d(\cdot, G)$. Nevertheless, this function (or the function $\rho$) is often a useful tool for studying curves and higher dimensional surfaces.