Results in this paper improve upon results previously obtained by other authors on co-absolutes of \(\beta N-N\) \((=N^*)\) and Parovichenko spaces. A Parovichenko space is a compact, zero-dimensional space of \(\pi\)-weight c, with no isolated points, such that disjoint \(F_{\sigma}\)-sets have disjoint closures and every non-empty \(G_{\delta}\)-set has non-empty interior. The space \(N^*\) is the remainder of the Stone-Čech compactification of the positive integers (N). Two spaces are called co-absolute if their extremally disconnected projective covers are homeomorphic (or equivalently, if their Boolean algebras of regular closed sets are isomorphic). Results by other authors published previous to the publication of this paper have shown that (i) CH (continuum hypothesis) implies that every Parovichenko space is co- absolute with \(N^*\), and (ii) If every Parovichenko space (even of weight c) is co-absolute with \(N^*\) then \(c<2^{\omega_ 1}\). It is shown that the converses to each of (i) and (ii) are false.