For each $s\in\{2,4\}$, the generating function of $R_s(n)$, the number of partitions of $n$ into odd parts or congruent to $0$, $\pm s\pmod {10}$, arises naturally in regime III of Rodney Baxter's solution of the hard-hexagon model of 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 of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. In this paper, we investigate the parity of $R_s(n)$ and $R^*_s(n)$, providing new parity results involving sums of partition numbers $p(n)$ and squares in arithmetic progressions.