  A holomorphic curve in moduli spaces is the image of a non-constant
holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli
space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that,
when all peripheral monodromies are of infinite order, the holomorphic map is a
quasi-isometric immersion with parameters depending only on $g$, $n$, $h$ and
the systole of $B$. When peripheral monodromies also satisfy an additional
condition, we find a lift quasi-isometrically embedding a fundamental polygon
of the hyperbolic surface $B$ into the Teichm\"uller space. We further improve
the Parshin-Arakelov finiteness theorem, by proving that there are only
finitely many monodromy homomorphisms induced by holomorphic curves of type
$(g,n)$ in $\mathcal{M}_h$ where systole is bounded away from $0$, up to
equivalence.
