I know that for a circle, the parametric equation is $(x(t),y(t))=(\sin(t+a),\cos(t+a+\pi n))$ where $n\in\mathbb Z$ and $a\in\mathbb R$.
I was trying to make work for a sphere, which would only cover a portion of it, and I think I got a particular solution $$(x(t),y(t),z(t))=\big(\cos(\sin(et))\sin (t),\cos(\sin(et))\cos (t),\sin(et)\big)$$
I know that $e$ can be replaced with nearly any irrational number, and I think $\sin(t)$ and $\cos(t)$ can be replace in a similar way the circle one can, but I'm not sure. I don't see what else can be generalized, but I know that it definitely can be generalized further.