  We prove that Zhu's algebra and the $C_2$-algebra of type ${\tt C}_m$ have
the same dimension, and we compute the graded character of the latter. Maximal
parabolic subalgebras of the symplectic algebra play a central role in our
construction. For the orthogonal algebras our methods do not allow to describe
the whole $C_2$-algebras, we get only a description of a certain quotient of
the algebra.
