I figured I could improve on the first answer I gave; some more tweaking with the $z$-component led me to the parametric equations
$\begin{align*}x&=v|\cos\,u|^p \cos\,u\\y&=v|\sin\,u|^p \sin\,u\\z&=\frac{hv}{c} \left(\left(\frac{v}{c}+2f-2\right)\cos^2 2u-2f\right)\end{align*}$
where $f > 0$ is an additional adjustable parameter.
Here is the case $p=1$, $c=9/10$, $h=1/2$, $f=2/3$, and $0 \leq v \leq 3/2$:

I derived these new equations by starting with the parametric equations of a cone, replacing the circular cross sections with Lamé curve cross sections, and tweaking the $z$-component such that the sweeping ray linearly interpolates between a line (between corners) and a parabola (on the corners).