Let G be a parahoric group scheme over a complex projective curve X of genus greater than one. Let G denote the moduli stack of G-torsors on X. We prove several results concerning the Hitchin map on TG. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that 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. © 2018 The Author(s).