In this article, we shall describe the center of the universal affine vertex superalgebra $V^{ \kappa_c}(\mathfrak g)$ associated with $\mathfrak g=\mathfrak{ sl}_{ 2|1}, \mathfrak { gl}_{ 2|1}$ at the critical level $\kappa_c$ and prove the conjecture of A. Molev and E. Ragoucy in this case. The center $\mathfrak{ z}(V^{ \kappa_c}(\mathfrak{ sl}_{ 2|1}))$ turns out to be isomorphic to the large level limit $\ell \rightarrow \infty$ of a vertex subalgebra, called the parafermion vertex algebra $K^{ \ell} (\mathfrak{ sl}_2)$, of the affine vertex algebra $V^\ell(\mathfrak{ sl}_2)$. The key ingredient of the proof is to understand the principal $W$-superalgebra $ W^{ \kappa_c}(\mathfrak{ sl}_{ 2|1})$ at the critical level. It relates the center $\mathfrak{ z}(V^{ \kappa_c}(\mathfrak{ sl} _{ 2|1}))$ to $V^\infty(\mathfrak{ sl}_2)$ via the Kazama-Suzuki duality while it has a surprising coincidence with $V^{ \kappa_c}(\mathfrak{ gl}_{ 1|1})$, whose center has been recently described. Moreover, the centers $\mathfrak{ z}(V^{ \kappa_c}(\mathfrak{ sl}_{ 2|1}))$ and $\mathfrak{ z}(W^{ \kappa_c}(\mathfrak{ sl}_{ 2|1}))$ are proven to coincide as a byproduct. A general conjecture is proposed which describes the center $\mathfrak{ z}(V^{ \kappa_c}(\mathfrak{ sl}_{ n|m}))$ with $n>m$ as a large level limit of ``the dual side'', i.e., the parafermion-type subalgebras of $W$-algebras $ W^\ell(\mathfrak{ sl}_{ n}, \mathbb{ O}_{ [n-m,1^m]})$ associated with hook-type partitions $[n-m,1^m]$, known also as vertex algebras at the corner.
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