I finally found an example of women having less rights than men in the West.

Apparently in Italy women don't have the right to be professional athletes. This right is granted to men but only in four disciplines: golf, basketball, bycicle racing and soccer. The rationale for this exclusion is that professional athletes have to pay higher taxes and that would affect negatively the budget of female sport teams that typically suffer from lower revenues and attract a lower number of fans compared to men's.
On a theoretical standpoint I recognize it's unfair but as you can see there's a practical reason for this "discrimination".