We can fill the $3\times3$ gray area in $2^9 = 512$ ways and then adjust the rows and columns with the yellow regions such that each row and columns eventually contains odd no of $1$'s.
Then observed the following:
Region A and B, the sum of bits in each of the regions must be odd. Therefore both yellow regions must be either odd or even at the same time. So, there is no conflict in the left bottom corner position. Finally, the answer is $512$.