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We study the truncation functors and show the existence of projective coverof each irreducible module in parabolic BGG category $\mathcal O$ over infiniterank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is aKoszul category. As a consequence, the corresponding parabolic BGG category$\overline{\mathcal O}$ over infinite rank Lie superalgebra of types$\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.

Projective modules over classical Lie algebras of infinite rank in the parabolic category