Is it permissible to inform one's husband about the details of the relationship between her parents, their problems, their temperaments, and some past secrets?

There is no benefit in these questions, and a wife should not answer her husband about them, especially if they relate to past sins and transgressions. This is because a Muslim is commanded to conceal their own faults and the faults of others. Disclosing sins could be a reason for them not to be concealed on the Day of Resurrection. The Prophet, peace and blessings be upon him, said: "All of my Ummah will be forgiven except for those who publicize their sins," and "Whoever conceals [the fault of] a Muslim, Allah will conceal his fault on the Day of Resurrection," and "Allah does not conceal a servant [from sins] in this world except that Allah will conceal him on the Day of Resurrection."