Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs?
Interesting question. Parallelograms are quadrilaterals for historical reasons. They could have been defined to include your examples, but weren't. Now the meaning is so common that it can't be changed.
I don't think there is a name for your class of polygons. The reason is in this:
It seems to me there must be some value in describing a polygon with even numbers of sides in which the opposing sides are parallel to each other.
If there were some value - if these polygons came up often in geometry - then someone would have named them. If you have interesting things to say about them and publish your thoughts you'll invent a name in your paper. If it's widely read the name will stick.
I thought parallelogon would be a good possibility, but that name is taken: https://en.wikipedia.org/wiki/Parallelogon .
The convex polygons whose sides come in equal parallel pairs are zonogons: https://en.wikipedia.org/wiki/Zonogon . Your polygons have zonogons as nontrivial Minkowski summands.
I'm going to propose, out of the blue, terms like "hexaparallelogram", "octaparallelogram", and so forth.
I'm wondering whether, for more than $4$ sides, you would like your definition of hexaparallelogram to be restricted to having 3 pairs of parallel and pairwise equal sides (as in your picture - evidently these have a name, zonogon), or would you include a hexagon with vertices at $\{(0,0), (12,0), (16,6), (4,12), (0,12), (-6,3)\}$ which has three pairs of parallel sides but no two sides of equal length?
Euclid, in proposition 34, introduces the term (παραλληλόγραμμα χωρία) which we can translate to "parallelogrammic area." So much for the etymology sites that trace the word only to Middle French. Euclid himself restricted the word to just four-sided figures. Proclus credits Euclid with having introduced the term "parallelogram," as opposed to bringing down that term from earlier works. So that tells us who to blame.