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The degree-zero polynomial is constant and therefore even. Assume inductively that $p_j(-x)=(-1{)}^j{p}_j(x)$ for $j<k$. Define $q_k(x)=(-1{)}^k{p}_k(-x)$, which has the same leading coefficient as $p_k$.

Changing variable $x\mapsto -x$ and using the even weight and inductive hypothesis gives $⟨q_k,p_j{⟩}_w=(-1{)}^{k+j}⟨p_k,p_j{⟩}_w=0$ for $j<k$. Hence $q_k-p_k$ has degree below $k$ and is orthogonal to every $p_0,\dots ,p_{k-1}$. Those polynomials form a basis for the polynomials of degree below $k$, so the difference is orthogonal to itself. Positivity of the weighted inner product forces it to be zero.

Thereforecompleting the induction and proving parity of orthogonal polynomials for an even weight. Thus even indices give even functions and odd indices give odd functions, regardless of the nonzero normalization chosen.