Hoi,
The parser would understand it because it stored information. The property is still the same property, the label it uses is now seen as a local overrride.
Daniel, there are many ways to solve this. The problem you face is based on a misconception. Language are not meant for rigidity. Expectting that you can has already been shown to be problematic. Consequently persisting on labels to be always unique is a problem of your own choosing. A problem that will not go away and is easiest solved now.
It is abundantly clear that you WILL use the requirement of Wikidata as an excuse when a language has no alternative. At that time it will be even more problematic to fix this issue. Not only because of the amount of data that may need conversion but also because assumption about Wikidata have grown even more fixed and rigid.