a) ab + cd is odd. thus one is even and other odd.
lets say cd is odd, then c,d -> odd. atleast one of a,b is even.
c/d is odd... also a/b can be even or odd (4/2, 4/4) thus insufficient
b) a/b + c/d = (ad+bc)/bd.
as ad+bc is odd, numerator is odd and thus a/b + c/d is odd.....
Sufficient
IMO B

Given A/B = Integer and C/D = Integer.
Is (AD+BC)/BD Odd or Even?

Statement 1
AB+CD = Odd. Either AB is Even and CD is Odd or AB is Odd and CD is Even. As Even +Odd = Odd

If AB is Even, then A and B are both Even, or either A or B are Even or Odd
If CD is Even, then C and D are both Even, or either C or D are Even or Odd

Even/Even = Even or Odd. Even/Odd = Even. So the result could be either Even or Odd. Insufficient

Statement 2
AD+BC= Odd. Substitute this into the main prompt. Is Odd/BC Even or Odd. If the numerator is Odd, then BC has to be Odd, otherwise the result will never be an integer. Sufficient

I like the approach you both have taken to Statement 2, but in case others didn't think of it that way, as I didn't, another approach to statement 2:

IF A,B,C AND D ARE INTEGERS. AND A/B AND C/D ARE INTEGERS. IS A/B + C/D ODD OR EVEN ?
II: AD + BC IS ODD

AD + BC must be E + O. Let's assume AD is the odd one, since the situation is the same either way.
Then, A is odd, D is odd, and at least one of B or C is even.
So, the numbers are either OOOE or OOEE

Now, if it were OOEE, either B or D would have to be even, which would mean that in the sum A/B + C/D we would be doing O/E in one of the terms, which would be impossible.

So, it must be OOOE, with either A or C as the sole even number.
So, A/B + C/D = E/O + O/O.

As others have noted, O/O = O and E/O = E, so the sum A/B + C/D = O.