  This paper is about the orbifold theory of affine and parafermion vertex
operator algebras. It is known that the parafermion vertex operator algebra
$K(sl_2,k)$ associated to the integrable highest weight modules for the affine
Kac-Moody algebra $A_1^{(1)}$ is the building block of the general parafermion
vertex operator $K(\mathfrak{g},k)$ for any finite dimensional simple Lie
algebra $\mathfrak{g}$ and any positive integer $k$. We first classify the
irreducible modules of $\mathbb{Z}_{2}$-orbifold of the simple affine vertex
operator algebra of type $A_1^{(1)}$ and determine their fusion rules. Then we
study the representations of the $\mathbb{Z}_{2}$-orbifold of the parafermion
vertex operator algebra $K(sl_2,k)$, we give the quantum dimensions, and more
technically, fusion rules for the $\mathbb{Z}_{2}$-orbifold of the parafermion
vertex operator algebra $K(sl_2,k)$ are completely determined.
