Parity of two numbers if given parity of one of them and their binomal coefficient

Question

If we are given that $n$ is odd and for some $1 \leq k \leq n-1$ we have $n \choose k$ is even. Can we conclude anything about the parity of $k$?

Answer 1

No, as you can see from $$ \binom{5}{2} = \binom{5}{3} = 10. $$