JavaScript (Node.js), 57 bytes

for(a='x=>',i=0;a+=i+'==x?x%2?"odd":"even":',~i++;)_=>a+_

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tsh suggests bitwise operator.

JavaScript (Node.js), 57 bytes

for(a='x=>',i=0;a+=i+'==x?x%2?"odd":"even":',~i++;)_=>a+_

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tsh suggests bitwise operator.

Python 3.8 (pre-release), 67 bytes

print("def f(n):n in",[*range(2**32)],"==print('eovdedn'[n&1::2])")

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pog

JavaScript (Node.js), 64 bytes

_=>a+_
for(a='x=>',i=0;i<2**32;)a+=`x==${i++}?x%2?"odd":"even":`

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.

JavaScript (Node.js), 57 bytes

for(a='x=>',i=0;a+=i+'==x?x%2?"odd":"even":',~i++;)_=>a+_

Try it online!

tsh suggests bitwise operator.