Let G be a connected reductive group defined over a local non Archimedean field F with residue field F; let P be a parahoric subgroup with associated reductive quotient M. If σ is an irreducible cuspidal representation of M(F) it provides an irreducible representation of P by inflation. We show that the pair (P,σ) is an ${\mathfrak S}$-type as defined by Bushnell and Kutzko. The cardinality of${\mathfrak S}$ can be bigger than one; we show that if one replaces P by the full centraliser ${\hat P}$ of the associated facet in the enlarged affine building of G, and σ by any irreducible smooth representation ${\hat σ}$ of ${\hat P}$ which contains σ on restriction then (${\hat P}$,${\hat σ}$) is an ${\mathfrak s}$-type for a singleton set ${\mathfrak s}$. Our methods employ invertible elements in the associated Hecke algebra${\mathcal H}$ (σ) and they imply that the appropriate parabolic induction functor and its left adjoint can be realised algebraically via pullbacks from ring homomorphisms.