A reference for the "pro-unipotent radical" of a parahoric subgroup is given in
Moy, Allen; Prasad, Gopal; (1994). "Unrefined minimal K-types for p -adic groups." Inventiones Mathematicae 116(1): 393-408.
The definition is given at page 387. The authorauthors use the terminology "pro-nil radical" instead of "pro-unipotent radical", but this is the same object.