  We prove that the moduli space ${\mathcal A}_{g,\Gamma_0(p)}\otimes \bar
{\mathbb F}_p$ of principally polarized abelian varieties of dimension $g$ with
a $\Gamma_0(p)$-level structure in characteristic $p$ has $2^g$ irreducible
components.The cases of other parahoric level structures are also considered.
