The equation of a circle centered on the x-y plane is $ x^2 + y^2 = r^2 $.
Is it possible to determine a similar parametric equation for a centered circle raised/lowered by $z$ (i.e. parallel to x-y plane) with the condition $ x^2 + y^2 + z^2= r^2 $?
If the circle inside the sphere has height $z$, then its radius is $\sqrt{r^2-z^2}$ (make a drawing), and the parametrization you want is simply $$t \mapsto (\sqrt{r^2-z^2}\cos t, \sqrt{r^2-z^2}\sin t, z).$$