Parity of binomial coefficients

Lucas mod 2. $\binom{n}{k}\equiv1\pmod2$ iff every binary digit of $k$ is $\le$ the corresponding digit of $n$.

Equivalent. The binary expansion of $k$ is a submask of that of $n$.

Conclusion. $\binom{n}{k}$ is odd iff no bit of $k$ exceeds the corresponding bit of $n$ in binary (i.e. $k\,\&\,n=k$ in bitwise AND).