ISL 2022 A7

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=$ $x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ be a polynomial, where $n \geqslant 2$ and $a_{i}$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k, s(k)$ and $s(P(k))$ have the same parity? (Belarus)