In this paper, the author mentions before Theorem 23 that the result was proven by Elmo Moore in 1968; the associated reference says "unpublished result". I would be grateful to anyone who knows a reference to the result (or simply a proof thereof).
For reference, this theorem reads:
Let $p_1(n)$ be the number of partitions of $n$ into parts which are either divisible by $4$ or odd. Let $q_1(n)$ denote the number of partitions of $n$ of the form $n_1+n_2+\cdots + n_s$, where $n_i\ge n_{i+1}$ and, if $n_i$ is odd, $n_i-n_{i+1}\ge 1$ for $1\le i\le s$. Then $p_1(n) = q_1(n)$.