  We compute the relative orbifold Gromov-Witten invariants of
$[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$, with respect to vertical
fibers. Via a vanishing property of the Hurwitz-Hodge bundle, 2-point rubber
invariants are calculated explicitly using Pixton's formula for the double
ramification cycle, and the orbifold quantum Riemann-Roch. As a result parallel
to its crepant resolution counterpart for $\mathcal{A}_n$, the GW/DT/Hilb/Sym
correspondence is established for $[\mathbb{C}^2/\mathbb{Z}_{n+1}]$. The
computation also implies the crepant resolution conjecture for relative
orbifold Gromov-Witten theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times
\mathbb{P}^1$.
