  There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n)
\hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) =
\text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural
generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called
the algebra of generalized parafermions by the third author and was shown to
arise as a one-parameter quotient of the universal two-parameter
$\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper,
we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n)
= \text{Com}(V^k(\mathfrak{so}_{2n}),
V^k(\mathfrak{so}_{2n+1}))^{\mathbb{Z}_2}$. We realize this algebra as a
one-parameter quotient of the two-parameter even spin
$\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,4,\dots)$, and we
classify all coincidences between its simple quotient $\mathcal{D}_k(n)$ and
the algebras $\mathcal{W}_{\ell}(\mathfrak{so}_{2m+1})$ and
$\mathcal{W}_{\ell}(\mathfrak{so}_{2m})^{\mathbb{Z}_2}$. As a corollary, we
show that for the admissible levels $k = -(2n-2) + \frac{1}{2} (2 n + 2 m -1)$
for $\widehat{\mathfrak{so}}_{2n}$ the simple affine algebra
$L_k(\mathfrak{so}_{2n})$ embeds in $L_k(\mathfrak{so}_{2n+1})$, and the coset
is strongly rational. As a consequence, the category of ordinary modules of
$L_k(\mathfrak{so}_{2n+1})$ at such a level is a braided fusion category.
