Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions $C(n)$ and $D(n)$ related to Euler's theorem. Lin and Zang extended their result to Glaisher's theorem by generalizing $C(n)$. We generalize $D(n)$ and prove an analogous partition identity for the $m=3$ case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases.