Given a parametric curve $\{f_x[t], f_y[t]\}$, the parametric formulas for its parallel (two branches) with a positive offset $d$ is
$$ \left\{f_x[t] + \dfrac{df_y'[t]}{\sqrt{f_x'[t]^2 + f_y'[t]^2}},\, f_y[t] - \dfrac{d f_x'[t]}{\sqrt{f_x'[t]^2 + f_y'[t]^2}} \right\} $$
[Here is a picture showing an example of a parallel curve of a sine wave.]
Is it possible to find a curve parallel to a curve but without these loopy "cusp" things on top? It would look like the top curve in the picture, but with the parts cut off where it crosses itself.
The red curve ( that appears as a sine curve) has a wavelength/period, say $p$ and amplitude $A$. For cosine curve $ y= A \cos \dfrac{2 \pi x}{\lambda} $ we can find minimum radii of curvatures. Radius of curvature at crest or trough is found by differentiation using standard curvature formula to be:
$$ R= \dfrac{ \left( {\lambda/{2 \pi}} \right)^2}{ A} $$
The given green and purple curves are here drawn with $ d \approx \lambda/2 >R $.
If they had been drawn with $d<R,$ no cusp will appear and the parallel curve would be smooth and wavy.
If they had been drawn exactly with $d= R $ one cusp only will appear. What forms in this case "touches" evolute normally by contact at the vertex center of curvature.
And when $d>R$ there would be two cusps with ever increasing domes and troughs of the parallel curves coverage with crests and troughs producing bigger arcs spanning between their bi-cusps.
Parallel Curve Wiki first figure illustrates it all, with single minimum it is less complicated. They are offset curves in Engineering parlance, and for 3-space they are known as Bertrand parallel Surfaces.