Affine Brauer category and parabolic category $\mathcal O$ in types $B, C, D$

A strict monoidal category referred to as affine Brauer category $\mathcal{AB}$ is introduced over a commutative ring $\kappa$ containing multiplicative identity $1$ and invertible element $2$. We prove that morphism spaces in $\mathcal{AB}$ are free over $\kappa$. The cyclotomic (or level $k$) Brauer category $\mathcal{CB}^f(\omega)$ is a quotient category of $\mathcal{AB}$. We prove that any morphism space in $\mathcal{CB}^f(\omega)$ is free over $\kappa$ with maximal rank if and only if the $\mathbf u$-admissible condition holds in the sense of (1.30). Affine Nazarov-Wenzl algebras and cyclotomic Nazarov-Wenzl algebras will be realized as certain endomorphism algebras in $\mathcal{AB}$ and $\mathcal{CB}^f(\omega)$, respectively. We will establish higher Schur-Weyl duality between cyclotomic Nazarov-Wenzl algebras and parabolic BGG categories $\mathcal O$ associated to symplectic and orthogonal Lie algebras over the complex field $\mathbb C$. This enables us to use standard arguments in [1,26,27] to compute decomposition matrices of cyclotomic Nazarov-Wenzl algebras. The level two case was considered by Ehrig and Stroppel in [14].