$\mathsf{co\text{-}NP}$ and Cook reductions
Every NP-complete program $A$ is co-NP hard under Cook reductions: given a problem $B$ in co-NP, its complement $\overline{B}$ is in NP, so there is a polytime function $f$ such that $f(x) \in A$ iff $x \in \overline{B}$. Therefore the following is a Cook reduction from $B$ to $A$: given $x$, ask whether $f(x) \in A$, and return the opposite.

This shows that Cook reductions are not sensitive to the difference between NP and co-NP. Karp reductions are, and that's why we use them in the usual definition of NP-complete. (If we were only interested in P vs. NP, Cook reductions would be fine.)