The article contains 36 occurences of the word 'collapse' but this certainly does not mean MWI and Collapse are the same interpretation.

The article's own use of 'parallel worlds' appears to mean 'any world other than the one we currently occupy so "X is parallel to Y" means "X and Y both exist and X =! Y".

Using this definition we can answer your question quite easily, if W1 is parallel to W2 and W2 is parallel or W3 but W1 is not parallel to W3, we can deduce that W1 = W3, and so this is what 'second order parallel' means, identical.

You see what I mean when I say that treating parallel as a relation on worlds is a pretty vacuous way of defining it. Essentially you have drawn the complete graph of worlds, which in terms of information is pretty much equivalent to the empty graph of worlds.

Parallel worlds is the same thing as the MWI.

This is just not true. There are all sorts of ways you can have parallel worlds.