Note that Example 4.3.5 in Rieffel and Polak designs a similar measurement for a two-qubit system. Expanding this to three qubits, we can write the Hermitian operator
(1)
\begin{align} \hat{O} = 2\Big(\ket{000}\bra{000} + \ket{011}\bra{011} + \ket{101}\bra{101} + \ket{110}\bra{110}\Big) + 3\Big(\ket{001}\bra{001} + \ket{010}\bra{010} + \ket{100}\bra{100} + \ket{111}\bra{111}\Big) \end{align}
which specifies measurement with respect to the decomposition of the vector space into subspaces $S_\textrm{even}$ and $S_\textrm{odd}$, which are generated by $\{ \ket{000},\ket{011},\ket{101},\ket{110} \}$ and $\{ \ket{001},\ket{010},\ket{100},\ket{111} \}$, respectively, where we have (arbitrarily) assigned the eigenvalue 2 to “even” and the eigenvalue 3 to “odd”.