Take Japanese writing for example, you definitely don't parse individual kana. Same with hangul, as you mention.
Even for Chinese writing it is true but to a very small extent, but my point was that it is much more important for phonetic writing, and most important indeed for alphabetic. I have even heard that it is less pronounced in Cyrillic, because all letters basically look like majuscles so the outline of words is quite uniform.
I don't quite get the comment about alphabetic script not being very phonetic, it's true that this would promote parsing words as units, but the main reason we do it is because it's much faster, not because some spellings are opaque. Kana spelling is almost completely one-to-one but it still makes sense to read e.g. -ma.shi.ta as a unit.
This is only really the case for phonetic writing where you learn to recognise the shape of groups of letters (this is how you are reading this text). For logographic/pictographic writing it is much less of a benefit.
And as I wrote above, even for alphabetic writing like Greek it doesn't really matter if all you read is the occasional monumental inscription.