For a positive integer \(n\) we denote by \(s(n)\) the sum of the digits of \(n\). Let \(P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\) 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?