A Bijective Proof for the Parity of Stirling Numbers

We give a bijective proof for the identity \(S(n,k) \equiv \binom{n-j-1}{n-k} \pmod{2}\)
where \(j = \lfloor \frac{k}{2} \rfloor\) is the largest integer \(\leq\frac{k}{2}\) .