From More Than Two where "parallel poly" is found: "This glossary is intended as a guide to many of the terms you might hear in the polyamorous community. Some of the terms have definitions that are not clearly established or universally accepted, particularly with regards to terms used to describe various relationship styles."
Given the above, and that historically, the importance of interrelationships are at the foundation of polyamory. Discarding interrelationships in polyamory is like discarding exclusivity in mono relationships. It makes no sense. I don't personally see how anyone can reasonably claim they are in a polyamorous relationship if there are no interrelationships.
To me it seems more like someone decided to make-up the term "parallel poly" to make it seem more acceptable to those who like to think of themselves as poly, but don't fit the criteria. This seems to happen a lot out there, especially on dating sites, where people I wouldn't think of as poly in a million years label themselves as poly just to get access to additional partners.
Personally, I don't have any moral problem with open relationships or swinging, so it's fine with me if that's what people are into. But you can count me as being on the other side of this particular issue where labeling is concerned. I trust that is okay. Or does everyone here have to agree all the time?