In "Marriage is a Private Affair" by Chinua Achebe, how do we see children raised by extremely strict parents becoming rebellious?

The issue here is complicated by cultural context. The studies of authoritarian parenting done in the United States are situated within a sophisticated modern society, and focus on middle class Caucasians among whom arranged marriages are regarded as very alien and excessively authoritarian. The Ibo society of Nigeria portrayed by Achebe is not the United States in the twenty-first century and different cultural norms apply.
The key distinction Achebe is making is between traditional village life and an urban life that is postcolonial, having adapted many of the values of British society. The young couple in the story is part of a transitional population with ties to both worlds, educated and sophisticated, but still with emotional connections to the traditions of extended family and village culture.
Nnaemeka's father is not portrayed as overly controlling in terms of Nigerian village society, and in fact the reactions of the villagers show his ideas and behaviors to be relatively normal, even if his continued refusal to accept Nene eventually appears overly stubborn. Nnaemeka is not really a rebel or juvenile delinquent, but simply a young man in love who desires his father's approval for his marriage. While he does marry the woman of his choice, and reacts to his father's authority by standing his ground, he is not rebellious in the sense of being driven to anti-social behavior out of resentment of authority. 
The story, in fact, ends with reconciliation due to the loving persistence of Nene. It is the father, not the son, who has maintained the rift, and even the father is eventually reconciled to the marriage by the existence of grandchildren. Overall, the arc of the story suggests that the title is ironic. Although marriage might on the surface appear private, in fact, it does involve the whole village and extended family. The story therefore is not simply, as it would be if set in modern Britain or the United States among middle class families, one of personal choices or rebellions but rather one about a clash of traditional Ibo and modern postcolonial Nigerian culture.