Should the Worker work less than the Stay At Home Parent at domestic duties?

Scenario-

A works outside the home between 8-12 hours a day, 5 days a week. A makes enough money for his wife, B, to stay at home with two small children. B is responsible for all domestic duties. All cleaning. All child rearing. A helps “watch” the kids when he is home so she can shower, but that is the extent of it. A does not clean, do laundry, cook, put the kids to bed, or fuss much with the kids at all, etc.

Since he works outside of the home, should he still have to contribute to household duties? Why or why not?