Yes, 1+2=3 and since 1 is odd and 2 is even this shows it can happen.
In fact it always does!
If we let an even number be denoted by 2n for some integer n and
and an odd one 2m+1 for some integer m.
Then, 2m+1+2n=2(m+n)+1. Call m+n, p since it is also an integer.
2p+1 is always odd.
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The sum of two odd numbers is always even.
The sum of one odd number and one even number is another odd number.
The sum of two odd numbers is an even number.
Because the sum of two odd numbers is always even and the sum of that even number plus another odd number is always odd.
No odd number of odd numbers can sum to an even number.