Did the design of armor in medieval China, Vietnam and Korea parallel armor in medieval Europe? In what ways were they the same? How were they different?

They are the parallel in that they are all variations on the Cataphract, a war technology that arose in Asia Minor.

So the armor from East Asia did indeed in some respects parallel armor in medieval Europe, in so far as it kept to the formula of 1) Armored horses that had been bred for that use, 2) armored rider wearing some form of scale armor, and 3) some form of lance. One of my own professors who was an expert in ancient warfare held to the theory that it was this technology and the economic, cultural, and power relationships (in the form of kings and lords, which we commonly associate with feudalism) that supported said technology was what defined the feudal era. Though I myself hesitate to present this as the fact of the matter, I have found it useful to refer to the various forms of "knights" in the various feudal periods of the Eastern Hemisphere as being variations of a cataphract.

They are different in that though they seem to all share this commonality, it is not clear if they are separate concurrent developments or all descended from the type of armored calvary seen on the Central Asian steppes. Therefore you see Knights in Europe wearing chain or plate, while the cataphracts of Japan were wearing heavier type plate at first, with a lighter armor involving small plates woven in with leather to increase flexibility to better accommodate a more infantry style of fighting and for mounted archery.

There are also differences in the culture that arose around the users of this technology, the "knights" themselves, so that though "knighthood" may look similar due to the economic necessities of maintaining mounted and armored knights, the bonds of loyalty and administrative/logistical organization may be completely different.