Peculiarly, neither ever asked "Whose job is this?" or asserted "That is not my responsibility!" (Para. 13)
Meaning: Strangely, neither the husband nor the wife asked "Whose job is this?" or declared forcefully "That is not my responsibility!"
Notice that this sentence describes the couple as caring and helpful to each other as they did not give their responsibilities to the other.
peculiarly: ad. 1) strangely
Peculiarly, the dog has three legs. 奇怪的是，这条狗只有三条腿。
Please seize the man. He has been hanging around my store all day and acting peculiarly. 请把那个人抓起来。他在我店四周游逛了一整天，行为异样。
2) especially
This book is peculiarly difficult. 这本书特别难。
assert: vt. declare forcefully
Sandy's mother asserts that it's not healthy to eat standing up. 桑迪的母亲断言，站着吃饭不卫生。
The judge asserts that he is not at fault. 法官宣布他无罪。
responsibility: n. duty; condition or quality of being mature
I take full responsibility for this action. 我对这次行动负全部责任。
Her responsibility is to look after the children. 她的责任是照看孩子。