1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$).

2) To parameterize an elliptic curve $y^2=4x^3-g_2x-g_3$, we can use $(x,y)=(\wp(t),\wp'(t))$.

(I know how to prove (1) and (2), and that it is possible to parameterize curves of higher genus using the unit disc.)

Question: Is there a common explanation for why both parameterizations are of the form $(F(t),F'(t))$? (as opposed to $(F(t),G(t)$) Does this phenomenon generalize to curves of higher genus?

alwaystrue if you 'define' your parameter $t$ by $dt = dx/y$. In each case, this defines $t$ as a multivalued function on the Riemann surface (= algebraic curve) (minus the poles of the differential $dx/y$, of course), or, if you prefer, it defines $t$ on a cover of the Riemann surface. You might ask when this $t$ has some special properties, such as, for example, that $dx/y$ be of the second kind, or have no poles, or some such condition, which might be of interest. – Robert Bryant Aug 4 '12 at 14:48