  The parafermionic cosets $C_k = \mathrm{Com} (H, L_k(\mathfrak{sl}_2) )$ are
studied for negative admissible levels $k$, as are certain infinite-order
simple current extensions $B_k$ of $C_k$. Under the assumption that the tensor
theory considerations of Huang, Lepowsky and Zhang apply to $C_k$, all
irreducible $C_k$- and $B_k$-modules are obtained from those of
$L_k(\mathfrak{sl}_2)$, as are the Grothendieck fusion rules of these
irreducible modules. Notably, there are only finitely many irreducible
$B_k$-modules. The irreducible $C_k$- and $B_k$-characters are computed and the
latter are shown, when supplemented by pseudotraces, to carry a
finite-dimensional representation of the modular group. The natural conjecture
then is that the $B_k$ are $C_2$-cofinite vertex operator algebras.
