Here is a partial answer: the [Continuum Hypothesis implies that all Parovichenko spaces are soft-Parovichenko][1]; the proof is a bit long, so I put it in a PDF-file on my website.

Also I retract my claim in the comments that all compactifications with $\omega_1+1$ as a remainder are soft. It is true, in ZFC, that $\omega_1+1$ is soft-Parovichenko but "all compactifications with remainder $\omega_1+1$ are soft" is equivalent to $\mathfrak{t}>\omega_1$.


  [1]: http://fa.its.tudelft.nl/~hart/37/publications/the_papers/soft-compactifications.pdf