So, first of all I often like to recommend Learn me a Bitcoin for Bitcoin basics.

As we can see, going from private to public key is done through ECDSA. You can read more about the algorithm on Wikipedia. In general, one of the basic requirements of signature algorithms is that they don't leak anything. One such thing would be if it the private key is even or odd, since it would be halving the input space. This would be an instant K.O. for any signature algorithm and nobody would use it.