The problem with your question is that we don't know how $S$ is distributed. Are the integers uniform, and if so within what range? Or are they distributed according to another probability distribution?
In truth, because you specify that all elements of $S$ are integers, we don't really need the entire probability distribution, just: the number $n_o = |S_o|$ of integers which are odd, and the size of the set $n=|S|$...
Then, supposing that you are sampling the elements without replacement (i.e., you cannot take the same element twice), then your probability of the sum being odd, is the probability of drawing an odd number of odd numbers. Thus it's the sum on all odd values of a hypergeometric distribution.
UPDATE: here are more details and a nice illustration.
We are defining an hypergeometric random variable $O$, the total size of the population is $n$ (the size of the set), the size of successes is $n_o$, the size of our sample is $m$ (what you call $N$).
The probability of being odd is:
$\sum_{i=0}^n \mathbb{P}[O=2i+1]=\sum_{i=0} \frac{\binom{n_o}{2i+1}\binom{n-n_o}{m-(2i+1)}}{\binom{n}{m}}$
(where the binomial coefficient is zero where there parameters are non-sensical :-)
Interestingly there is a regime where this probability is $1/2$ but it is not necessarily so, as illustrated by the following example. Take a sample of size $n=100$. Below is the plot of the probability of having an odd sum from drawing $m$ integers ($m$ is the $x$-axis, and the probability is the $y$-axis). As time goes by, we add more and more odd numbers to the set (first there are none, so the probability is always 0 then there is 1 so the probability of the sum being odd is equal to the probability of drawing that number, etc.).
As all integers are odd, then the probability of having an odd sum (since all integers are odd) depends deterministically on the size $m$ of the sample, hence the final saw-shaped plot.

(Plot of the probability that the sum of $m$ integers taken from $S$ is odd. The size of $S$ is $n=100$; $m$ varies on the $x$-axis from $0$ to $100$; each successive plot is $n_o$, the number of odd elements in the set $S$, varying from $0$ to $100$.)
MINOR EDIT: I accidentally called $n_e$ the number of odd elements... changed that to $n_o$.