Let's start considering the top-left $19\times 13$ square grid. They can be freely colored. Afterwards, we notice the that the first $13$ columns of the $20^{th}$ row are uniquely determined in order to make the number of orange squares in their corresponding columns even. Meanwhile, the last columns of the first $19$ rows are also uniquely determined by a similar reasoning. This left the right bottom corner square the only one to be determined.

Because the odd-even parity of the total numbers of orange squares in the initial $19\times 13$ grid is given. It will determined the odd-even parity of the first $13$ columns of the last row, and the first $19$ rows in the last column. Therefore, the color of the right bottom square can be uniquely determined without conflict.

This means that the number of ways is determined by the number of possible coloring schemes of the first $19\times 13$ grid, which is $2^{19\times 13}$. Hence, the answer is $19\times 13=\boxed{247}$.