We will associate an odd partition to every even partition. Let $(a_1,a_2,...,a_k)$ be an even partition, where $k\leq \frac{n}{2}$ as the $a_i$ are in non-decreasing order and $a_1\geq 2$. Hence the partition's odd associate will be $(1,1,1,...,1,a_1-1,a_2-1,...,a_k-1)$. This association is obviously unique, so if we want the number of odd and even partitions to be equal, all odd partitions must start with 1, and obviously n needs to be even, as for n odd there is no even partition. If $n\geq 6$, then an odd partition would be $(3,n-3)$ which can't be possible in the context of our problem. Hence, only $n=2$ and $n=4$ can be solutions, and it can be easily checked that these are both solutions.