  This article constructs the Satake parameter for any irreducible smooth
$J$-spherical representation of a $p$-adic group, where $J$ is any parahoric
subgroup. This parametrizes such representations when $J$ is a special maximal
parahoric subgroup. The main novelty is for groups which are not quasi-split,
and the construction should play a role in formulating a geometric Satake
isomorphism for such groups over local function fields.
