Are you familiar with generating functions?
Let be the number of ways to partition into at most odd parts and the number of partitions of such that each even part occurs an even number of times, respectively. Define to be the respective generating functions for , then
Let , then
where is the number of partitions of where the number of even parts is even. The number of ways to do this when is the coefficient of .
If not...I highly recommend generatingfunctionology by Wilf because it's a good book, but better yet, it's free to download at the link I provided. In particular, page 100 begins his discussion of using generating functions for partitions.
Hope this helps.