Imagine that the Roman Empire never fell. Imagine that they survived and thrived. Imagine that their traditions, customs, and laws never changed - as how things were then, that is how they are now.
In this system, the oldest male member of your family in sound health is now in charge. He would be the lawyer for the entire family, he would be the representative / senator of the entire family, he would be the businessman of the entire family, and he would be the priest of the entire family. He would be charged with the well-being of everybody under his care - from his retired parents to his newborn-children, to his unmarried daughters, to his sons and their wives and their children, and to the slaves that keep everything working. It was not uncommon for a Roman household to consist of several dozen members.
Paul could have affirmed the Roman household codes: Wives obey their Husband, Slaves obey their master, Children obey their Parents - but he didn't. Not in their entirety. He realized that while the system looks like it works, it's not God's way of doing things to 'do as the Romans do'. Paul could have out-right rejected the household codes as based on human principles and not what God had in mind for all people, for all times, or for all places. But he knew that the Romans were still in charge - and reading his letters before delivering them from prison.
John too, had the same problem. When he wrote Revelation, he used metaphor and poetry to cloak the real message that he was sending. Paul, on the other hand, couldn't use metaphor or poetry to set the record straight. He had to write exactly what the Romans would accept - 'follow the household codes' - with a twist - 'but be Christians about it.'
That makes his advice make sense - with Dad being the lawyer / priest / senator / representative / businessman rolled into one - he was a unit of the state and it was necessary for the family to make his job easier by not breaking out into a quarrel at the drop of a hat and by not making him and his family into a public disgrace. Paul was all about not making Christianity any more of a target than it already was - not standing out - not challenging Roman authority. But he didn't believe that the Roman way was God's way.
Now that the Roman Empire is long gone - Paul's advice remains the same: "follow American / Spanish / Russian / etc. standards for families, but be Christians about it."
We're supposed to look like those around us in terms of being a family - but we're to act significantly different. So when it comes to the household codes - we should not just follow them because they're in the Bible - they weren't written for us who don't live according to the Roman household codes. But we can still learn from them.