Why does LEGO state the number of parts on US boxes, but not in Europe?
There is indeed a requirement that packaged products sold in the US must indicate the weight, volume or quantity plainly on the package. This is regulated by the FTC, which says in its requirements,


  The net quantity of contents (in terms of weight or mass, measure, or
  numerical count) shall be separately and accurately stated...


Canada has a similar law that says the net quantity of packaged products must be on the label.

The European Union's product labeling requirements do not appear to make the quantity a mandatory part of the product label. I do not think it actually prohibits putting the number of pieces on the packages, so it may not explain why The LEGO Company leaves that information off their European packages.