Formula for curve parallel to a parabola

I'll use the parametrization

$$\begin{align*}x&=2at\\y&=at^2\end{align*}$$

where $a$ is the focal length (the distance from vertex to focus).

Using the formula for a parallel curve of $(f(t)\quad g(t))^T$ at a distance $c$:

$$\begin{pmatrix}f(t)\\g(t)\end{pmatrix}+\frac{c}{\sqrt{f^\prime(t)^2+g^\prime(t)^2}}\begin{pmatrix}g^\prime(t)\\-f^\prime(t)\end{pmatrix}$$

we find the parametric equations

$$\begin{align*}x&=2at+\frac{ct}{\sqrt{1+t^2}}\\y&=at^2-\frac{c}{\sqrt{1+t^2}}\end{align*}$$

The corresponding Cartesian equation is rather complicated:

$$\begin{align*}x^2 \left(-8 a^3 y+x^2 \left(a^2-10 a y-3 c^2+y^2\right)-20 a^2 c^2+32 a^2 y^2+2 a c^2 y-8 a y^3+3 c^4-2 c^2 y^2+x^4\right)&=\\(c-y) (c+y) \left(4 a(a-y)+c^2\right)^2\end{align*}$$

so you're better off sticking to a parametrization.

Here's a plot of a bunch of parabola parallels: