To find the equivalent regular expression, try this approach.
If $\#a$ and $\#b$ are both odd then the total length of the string is even. Divide the string into two character blocks. Each block is either $aa$, $bb$, $ab$ or $ba$.
$aa$ and $bb$ have no effect of the parity of $\#a$ and $\#b$ so they are equivalent as far as our regular expression is concerned. Similarly $ab$ and $ba$ both change the parity of $\#a$ and the parity of $\#b$ so they are also equivalent.
So our regular expression will consist of blocks like $(aa+bb)$ and $(ab+ba)$.
We don't care how many $(aa+bb)$ blocks we see, but we need to see at least one $(ab+ba)$ block, so our regular expression will start with
After this we need to see an even number of $(ab+ba)$ blocks (remember zero is an even number), possibly interspersed with zero or more $(aa+bb)$ blocks. I'll let you take it from there.