  Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a
surjective morphism $\phi_E: X_0(N) \to E$ defined over $\mathbb{Q}$. In this
article, we discuss the growth of $\mathrm{deg}(\phi_E)$ and shed some light on
Watkins's conjecture, which predicts $2^{\mathrm{rank}(E(\mathbb{Q}))} \mid
\mathrm{deg}(\phi_E)$. Moreover, for any elliptic curve over $\mathbb{F}_q(T)$,
we have an analogous modular parametrization relating to the Drinfeld modular
curves. In this case, we also discuss growth and the divisibility properties.
