Let $k$ be a non-archimedean locally compact field and let $G$ be the set of $k$-points of a connected reductive group defined over $k$. Let $W$ be the relative Weyl group of $G$, and let $\Hecke (G,B)$ be the Hecke algebra of $G$ with respect to an Iwahori subgroup $B$ of $G$. We compute the effects of $\Hecke (G,B)$ and $W$ on the $B$-fixed vectors of an unramified principal series representation $I$ of $G$. We use this computation to determine the dimension of the space of $K$-fixed vectors in $I$, where $K$ is a parahoric subgroup of $G$.