  Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve
$X$ of genus greater than one. Let $\mathrm{Bun}_{\mathcal{G}}$ denote the
moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results
concerning the Hitchin map on $T^*\!\mathrm{Bun}_{\mathcal{G}}$. We first show
that the parahoric analogue of the global nilpotent cone is isotropic and use
this to prove that $\mathrm{Bun}_{\mathcal{G}}$ is "very good" in the sense of
Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson
map whose generic fibres are abelian varieties. Together, these results imply
that the parahoric Hitchin map is a completely integrable system.
