Answer
I'm afraid the Bible doesn't speak of step relationships, and examples of people with children marrying are extremely rare. In the case of Mary and Joseph, Joseph adopted Jesus, so that he was not his step-father but his father. By the time Abraham remarried, Isaac and Ishmael were adults. In the case of polygamy, it appears that each wife was responsible only for her own children.
In ancient times, the father was the head of the household, and had authority over and responsibilities toward all who lived under his roof, whether they were near or distant relatives, servants or slaves. Households run by women were few and far between. If we can draw some correlation between the roles of husbands and wives, it would seem that the step-parent probably bears responsibilities toward the step-child similar to those of a natural parent as long as the child is living in the same household.
Answer by Ra McLaughlin
Ra McLaughlin is Vice President of Finance and Administration at Third Millennium Ministries.