Let $p (n|$distinct odd parts$)$ be the number of partitions of $n$ into distinct odd parts. Prove that $p(n)$ is odd if and only if $p(n|$distinct odd parts) is odd.
I know we're suppose to use the theorem on self-conjugate partitions...
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As @vadim123 have mentioned, this is a complete solution:
However, looking backwards at the "history" of yours, I sincerely recommend that following:
1, This is not CourseHero or Chegg, don't merely post the problem and expect us to solve it, we want to see that you have made progress!
2, Please put a "homework" tag on the question if this is your homework.
3, Start doing problems on your own, or you will not pass the course. Anyway, wouldn't the homework be easier if you understand the material?
If an involution acts on a set, the parity of the set is the same as the parity of the its fixed points under that involution. Apply that to conjugation acting on partitions of $n$ to see that the parity you are after is the same as that of the self-conjugate partitions.
Next try to see why the number of self-conjugate partitions of $n$ is equal to the number of partitions of $n$ into distinct odd parts. (You only need that they have the same parity, but the hint is that they actually are the same numbers.)