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For each $s\in\{2,4\}$, the generating function of $R_s(n)$, the number ofpartitions of $n$ into odd parts or congruent to $0$, $\pm s\pmod {10}$, arisesnaturally in regime III of Rodney Baxter's solution of the hard-hexagon modelof statistical mechanics. For each $s\in\{1,3\}$, the generating function of$R^*_s(n)$, the number of partitions of $n$ into parts not congruent to $0$,$\pm s\pmod {10}$ and $10-2s \pmod {20}$, arises naturally in regime IV ofRodney Baxter's solution of the hard-hexagon model of statistical mechanics. Inthis paper, we investigate the parity of $R_s(n)$ and $R^*_s(n)$, providing newparity results involving sums of partition numbers $p(n)$ and squares inarithmetic progressions.

$q$-Series congruences involving statistical mechanics partition functions in regime III and IV of Baxter's solution of the hard-hexagon model