That is correct, all prime numbers in $\mathbb Z$ which are greater than 2 are odd. Likewise all prime numbers less than $-2$ are also odd.
A prime number is a non-unit number (that is, not $-1$ or 1) which is divisible only by itself, the units and the associates of itself (the number times a unit). So, for example, $-7$ is prime, 14 is not.
And no, there is no wacky prime that can't be assumed odd. Except maybe $-2$ and 2. Those two are kind of wacky, in my opinion.
The thing is that in math (or "maths," if you prefer), if you haven't proven an assertion, that assertion might be false. If you make an assertion without proof, someone might challenge you to prove it.
Fortunately, the assertion that all primes greater than 2 are odd is an easy assertion to prove. So, if anyone challenges you on that, you can just quickly state the proof and move on.