I've solved the problem, which has to do with the generating function for $k$.

For example:

- $k = -108*t^2 + 36*t - 67$ always has even parity
- $k = -108*t^2 + 36*t - 7$  always has odd parity
- $k = -108*t^2 + 36*t - 84$ is $50/50$ odd and even

These, and other similar functions, generate nice groupings in residue
classes modulo 504 and explain the population distribution of odd and
even ranks in those residue classes.