  Let $f \colon X \to B$ be a complex elliptic surface and let $\DD \subset X$
be an integral divisor dominating $B$. It is well-known that the
Parshin-Arakelov theorem implies the Mordell conjecture over complex function
fields by a beautiful covering trick of Parshin. In this article, we construct
a similar map in the context of $(S, \DD)$-integral points on elliptic curves
over function fields to obtain a new proof of certain uniform finiteness
results on the number of $(S, \DD)$-integral points. A second new proof is also
given by establishing a uniform bound on the canonical height by means of the
tautological inequality. In particular, our construction provides certain
uniform quantitative informations on the set-theoretic intersection of curves
with the singular divisor in the compact moduli space of stable curves.
