What society says that? Even with the dreaded influence of feminism the majority of people still seem to view many roles as male and female.
Though the biggest difference is in how extreme people apply those roles. Some roles are more important to maintain, such as biblical standards for husbands and wives, etc.. but others are just culture based and while not vitally important may still have a psychological effect on a gender if they aren't following those roles. Either because they themselves feel they are lacking the ability to fulfill the role they 'should' be, or because of ridicule by others.
I'd say figure out what roles are important to maintain, and what roles you are ok with bypassing. Though, there is an Element of truth to what your dad says. If you marry a woman who's fine with being the one to take care of rodents, bugs, etc... problem solved. But if you marry a woman who can't handle that, then it Will become your responsibility. You will, after all, be the head of the household and ultimately if something needs to be handled it will be your responsibility. As a man (biblically speaking).