Let's quickly review some even/odd number rules.

even * even = even
even * odd = even
odd * odd = odd
even + even = even
even + odd = odd
odd + odd = even

For the product of integers M and N to be even, at least one of them must be even.

Statement 1

We'll deal with the sentences one at a time.

Sentence 1: With the exception of 2, all prime numbers are odd. However, we only know that ONE of our prime numbers is odd. So we could have (odd * odd) - (even * even) = odd - even = odd OR (odd * odd) - (odd * odd) = odd - odd = even. So N can be even or odd.

Sentence 2: Since 2 is even, 2P must be even, whether P is even or odd. This means 2P + 1 (which equals Q) must be odd. If Q is odd and P is either even or odd, M can be either even or odd.

Taken together, these sentences tell us that N can be even or odd and M can be even or odd. Insufficient.

Statement 2

Sentence 1: Since 2 is our only possible even prime number, and it is consecutive with only 3, which is 1 unit away, we know that both prime numbers must be odd. This gives (odd * odd) - (odd * odd) = odd - odd = even. So N is even.

At this point we don't need to look at the second sentence - whether M is even or odd, we know that M * N will be even if N is even. Sufficient.