Let $F$ be a $p$-adic field and $G$ be split over $F$ plus all other usual adjectives. Let $\pi$ be an admissible tempered representation with Iwahori-fixed vectors. Let $P\supset I$ be some parahoric subgroup of $G$. What is known about the circumstances for which $\pi$ has $P$-fixed vectors?

For example, principal series will admit fixed vectors under maximal parahorics, and the Steinberg representation of $G$ doesn't admit any $P$-fixed vectors for $P\supsetneq I$. What goes on in between?