  It is proved that the regularity of parafermion vertex operator algebras
associated to integrable highest weight modules for affine Kac-Moody algebra
A_1^{(1)} implies the C_2-cofiniteness of parafermion vertex operator algebras
associated to integrable highest weight modules for any affine Kac-Moody
algebra. In particular, the parafermion vertex operator algebra associated to
an integrable highest weight module of small level for any affine Kac-Moody
algebra is C_2-cofinite and has only finitely many irreducible modules. Also,
the parafermion vertex operator algebras with level 1 are determined
explicitly.
