We know that the $(x,y)\in \mathbb{R}^2$ such that
$$x=\cos^3 \theta, y=\sin^3 \theta$$
verify the equation. We'd like to impose a further restriction, i.e.
$$y^2+z^2= x$$
Hence,
$$\sigma(\theta)= (\cos^3 \theta, \sin^3 \theta, \sqrt{\cos^3 \theta - \sin^{6}\theta})$$
But this is only the upper half of the solid, right? And we'd have to restrict $\theta$.
I'm truly lost.
Is there some general guidelines as to how to find parametric equations in these cases?
There is symmetry with respect to $z=0$. So, separate parametrizations
Right half $z>0$ $$\sigma(\theta)= (\cos^3 \theta, \sin^3 \theta, +\sqrt{\cos^3 \theta - \sin^{6}\theta})$$
Left half $z<0$ $$\sigma(\theta)= (\cos^3 \theta, \sin^3 \theta, -\sqrt{\cos^3 \theta - \sin^{6}\theta}).$$
At $z=0$ check for repeated roots at golden ratio starter angles
$$ \cos \theta = \phi_1,\phi_2 $$