What is the difference between 勤め and 役目?
Both seems to mean duty (something that one is expected or required to do by moral or legal obligation), but what is the difference in their nuance and usage?
役目 is closer to "role" in meaning. 勤め is what the "role" requires you to do or what people expect you to do corresponding to the role.
You can call them "duty" as they are both roughly what you have to do.
役目 is usually more direct and specific than 勤め. On the other hand, 勤め sounds more abstract and takes one step further than 役目 in meaning. Basically,
b.) says: you have to take this role, because you are a parent.
d..) says: being a parent includes this as what you have to do.
To summarize, 勤め is used based on who you are as 役目 is used based on what others think.