  We give a recursive method for computing all values of a basis of Whittaker
functions for unramified principal series invariant under an Iwahori or
parahoric subgroup of a split reductive group $G$ over a nonarchimedean local
field $F$. Structures in the proof have surprising analogies to features of
certain solvable lattice models. In the case $G=\mathrm{GL}_r$ we show that
there exist solvable lattice models whose partition functions give precisely
all of these values. Here `solvable' means that the models have a family of
Yang-Baxter equations which imply, among other things, that their partition
functions satisfy the same recursions as those for Iwahori or parahoric
Whittaker functions. The R-matrices for these Yang-Baxter equations come from a
Drinfeld twist of the quantum group $U_q(\widehat{\mathfrak{gl}}(r|1))$, which
we then connect to the standard intertwining operators on the unramified
principal series.
  We use our results to connect Iwahori and parahoric Whittaker functions to
variations of Macdonald polynomials.
