  It was recently suggested that the su(N)_k+su(N)_p/su(N)_{k+p} coset
conformal field theories should be related to N=2 SU(N) gauge theories on
R^4/Z_p. In this paper we study various aspects of this proposal. We perform
explicit checks of the relation for (N,p)=(2,4), where the symmetry algebra of
the coset is the so called S_3 parafermion algebra. Even though the symmetry
algebra of the coset is unknown for generic (N,p) models, we manage to perform
non-trivial checks in the general case by using knowledge of the Kac
determinant of the coset CFT. We also find evidence that the conformal blocks
of the (N,p) model should factorise into a certain product of p (N,1) conformal
blocks. Precisely this structure is present in the instanton partition function
on R^4/Z_p.
