Missing Matter, Parallel Universes?

Yes and no. Your link refers to mirror matter based only on parity symmetry while I believe the paper at hand is more general. The arXiv preprint [arxiv.org] discusses this at the start:

Concerns about parity are irrelevant for our following discussions: they extend to a parallel sector (or sectors) of any chirality. Nevertheless, in the following we shall name the twin particles from the `primed' parallel sector as mirror particles.

To set things up, imagine stepping through a mirror and doing some physics experiments. You would expect everything to work out the same as before so long as "left" and "right" were reversed (...along the axis normal to the mirror...). That turns out not to be the case, which was surprising--some decades ago a few experiments with relatively exotic particles didn't work out as expected (brief history here [wikipedia.org]). Thus matter "through the mirror" and "before the mirror" are distinguishable. It's possible that matter through the mirror exists in our before-the-mirror universe, though it shouldn't interact much with the matter we're used to because the force-carrying particles need to be mirrored as well which ends up leaving only gravitational interactions. As you may have guessed, this is a potential candidate for dark matter. The lack of electromagnetic interactions would prevent distant mirror matter from being seen, and the lack of strong or weak interactions would nix many lab tests (like those that detect neutrinos, which are detected by their weak interactions).

My (poor) understanding of the paper is that they consider an essentially arbitrary parallel universe with wimpy interactions with our own universe (except gravitationally), not necessarily just one created by parity changes. In particular they focus on transitions of neutrons from our universe to the parallel one and use such transitions to explain an anomalous dependence on magnetic field direction in a previous experiment.

As usual, caution is the best plan. The authors call for more experiments, and I'm sure there are numerous explanations for their results that don't require (IMO) spooky transitions between parallel universes.

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