  The purpose of this work is to illustrate in a family of interesting examples
how to study the representation theory of vertex operator superalgebras by
combining the theory of vertex algebra extensions and modular forms.
  Let $L_k\left(\mathfrak{osp}(1 | 2)\right)$ be the simple affine vertex
operator superalgebra of $\mathfrak{osp}(1|2)$ at an admissible level $k$. We
use a Jacobi form decomposition to see that this is a vertex operator
superalgebra extension of $L_k(\mathfrak{sl}_2)\otimes \text{Vir}(p, (p+p')/2)$
where $k+3/2=p/(2p')$ and $\text{Vir}(u, v)$ denotes the regular Virasoro
vertex operator algebra of central charge $c=1-6(u-v)^2/(uv)$. Especially, for
a positive integer $k$, we get a regular vertex operator superalgebra and this
case is studied further.
  The interplay of the theory of vertex algebra extensions and modular data of
the vertex operator subalgebra allows us to classify all simple local
(untwisted) and Ramond twisted $L_k\left(\mathfrak{osp}(1 | 2)\right)$-modules
and to obtain their super fusion rules. The latter are obtained in a second way
from Verlinde's formula for vertex operator superalgebras. Finally, using again
the theory of vertex algebra extensions, we find all simple modules and their
fusion rules of the parafermionic coset $C_k = \text{Com}\left(V_L,
L_k\left(\mathfrak{osp}(1 | 2)\right)\right)$ where $V_L$ is the lattice vertex
operator algebra of the lattice $L=\sqrt{2k}\mathbb{Z}$.
