  In this paper we study the correspondence between the
$\hat{\textrm{su}}(n)_{k}\oplus
\hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ coset conformal field
theories and $\mathcal{N}=2$ SU(n) gauge theories on
$\mathbb{R}^{4}/\mathbb{Z}_{p}$. Namely we check the correspondence between the
SU(2) Nekrasov partition function on $\mathbb{R}^{4}/\mathbb{Z}_{4}$ and the
conformal blocks of the $S_{3}$ parafermion algebra (in $S$ and $D$ modules).
We find that they are equal up to the U(1)-factor as it was in all cases of
AGT-like relations. Studying the structure of the instanton partition function
on $\mathbb{R}^4/\mathbb{Z}_p$ we also find some evidence that this
correspondence with arbitrary $p$ takes place up to the U(1)-factor.
