The first square has odd parity, the second has even parity, etc. You may notice that they are all even numbers, except for 17. So the sum is odd. You don't have to add them up. If I replace 17 with 18, then the sum is even.
Addition preserves the parity.
Each domino covers one even and one odd number. 31 dominoes cover 31 evens and 31 odds, which sum to an odd number. So the two remaining red squares must sum to an odd number. So it cannot be done.