Many results in number theory start with "Let $p$ be an odd prime...", but it's rare to see something like "Let $p$ be a prime that is not divisible by $3$..." or similar. Why is that so, fundamentally?
Is it just psychology? Evenness and oddness are more familiar concepts than residues with arbitrary moduli, so statements involving them tend to be more concise, and thus are easily perceived as being more elegant and interesting.
Is it a consequence of the fact that mathematics is heavily biased towards $2$-ary operations, which align more naturally with a modulus of $2$ to yield results about the residue class of operands?
Or is there something else going on? It's hard to imagine how an explanation of the type "$\rm GF(2)$ is the only field that..." could be sufficient, because of course dozens of unique, potentially relevant properties can be found for every small prime. The fact that quadratic reciprocity only works for odd primes also doesn't seem like it would quite be enough to explain the colossal weight given to odd primes in number theory, compared to primes of any other residue class.
