Let $p_1(n,k)$ denote the number of partitions of $n$ with exactly $k$ parts in which each odd part is less than twice the smallest part and $p_2(n,k)$ the number of partitions of $n$ with exactly $k$ distinct parts in which each odd part is less than twice the smallest part. The authors prove that for $i=1$ or $2$, a positive integer $k$, a prime $p \geq k +i -1$, and $n > \alpha_i(k) p - k^{3-i}+(2-i)k$, $$ p_i(n,k) \equiv p_i(n-\alpha_i(k)p, k) \pmod{p}, $$ where $\alpha_1(k)$ is the least common multiple of 1 through $2k-1$ and $\alpha_2(k)$ is the least common multiple of the odd integers 1 through~$2k-1$.