Here's the process for solving:
First, we saw that the only number that could be in the top row middle square was a 9, because all the other odd numbers were already in that column. The 6 slotted in easily after, because the middle square already had a 6. The middle and middle-right 8's followed suit because of process of elimination (they couldn't be in any grey squares.) That left only a 2 to finish the middle column.
The right side filled out fairly easily because the middle row had 5 grey squares, meaning that each of them had to be odd, and each of the white squares had to be even. That allowed us to fill in the top right square and eliminate many of the other top row and right column numbers to be one of just a couple.
From here, the puzzle fell together pretty straightforwardly. Narrowing down those rows and columns from the previous image allowed us to narrow many of the grey squares down to just one or two numbers which, through process of elimination, gave us many of the even numbers across the top and bottom corners. From there, it was just hunting for the next straightforward "there's only one possibility left" over and over until the grid was filled. (Please let me know if you need further explanation, though! Really fun sudoku!