The integers from -5 to 3 inclusive are -5,-4,-3,-2,-1,0,1,2,3 - 9 in number.
The odd integers are -5,-3,-1,1,3 - 5 in number.
The even integers are -4,-2,0,2 - 4 in number.
Consider (A) first.
This is possible if both integers are even or both are odd.
Ways of selecting two odd integers is 5C2 = 10.
Ways of selecting two even integers is 4C2 = 6.
Or ways of selecting two integers such that their sum is even is 10+6 = 16.
Ways of selecting two integers from given 9 is 9C2 = 36.
Or required probability is 16/36.
Consider (B) next.
This is possible if one integer is odd and the other is even.
Number of ways this can happen is 5C1*4C1 = 20.
Or required probability is 20/36.
Consider (C) next.
This is possible if either both integers are even or one of the integers is even.
Ways of selecting both even integers is 4C2 = 6.
Ways of selecting one even integer is 5C1*4C1 = 20.
Or ways of selecting two integers such that their product is even is 26.
So the required probability is 26/36.
Consider (D) next.
Product of two integers is odd if both integers are odd.
Ways of selecting two odd integers is 5C2 = 10.
Or required probability is 10/36
Next consider (E)
The product is negative when one integer is negative and the other integer is not zero but positive.
Ways of selecting this is 5C1*3C1 = 15.
Or required probability is 15/36.

Among all the above (C) has the maximum probability which is 26/36 = 13/18.