The base field is a number field.

It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).

Is the converse also true?

Note that both conjectures are now theorems (Faltings).

Edit: To be clear, I'm referring to the Shafarevich conjecture for **curves**. That is, for any number field $K$, finite set of places $S$ of $K$ and integer $g > 1$, the set of curves over $K$ of genus $g$ with good reduction outside $S$ is finite.