Let $p(n)$ be the number of partitions of the positive integer $n$. What can we say about the parity of $p(n)$ if we know that the parity of the self-conjugate partitions is even?
Actually I manually computed $p(2)=2$ and $p(6)=11$ and both of them had an even number of self-conjugate partitions. So I think we can not generally discuss the parity of $p(n)$ with the given assumption. Is that right?
Hint: $$ p(n) = (\text{# self conjugate partitions})+(\text{# non-self conjugate partitions}) $$ You want to know the parity of $p(n)$, and you know that the number of self conjugate partitions is even.
What can you say about the parity of the number of non-self conjugate partitions? Think about how can you use the act of conjugation to determine this parity.