The child of never married parents followed the status of its mother. Under Roman law, only Roman marriage was relevant.* If the people in question were subject to local law and custom, local definitions presumably applied.** As I mentioned before, according to jurists in the Imperial period, a slave woman who was free at any point during her pregnancy gave birth to a freeborn child, even if she had been re-enslaved. It is doubtful, however, that this law was strictly enforced. The question of who had rights over the child was born followed similar, but even simpler logic: the child was in the potestas of the same person as its mother.
The status of the child of free parents who were married, on the other hand, seems on the surface to be a little bit more straightforward. The child would be freeborn and belonged in its father’s power. If, however, the father was not sui iuris (i.e. not in anyone else’s power) when the child was conceived, he did not have the same rights over it. Rather, the child was born into the potestas of the same person as its father, be that its grandfather or whoever had adopted its father.
When Roman law on pregnancy did not deal directly with the status of the child, it still held repercussions for it. When a marriage ended through divorce or the death of the husband and one party claimed that the former wife was pregnant, it was necessary to have some way of confirming or denying the assertion. She was to be examined by five midwives and the decision of the majority ruled. If she knew herself to be pregnant but didn’t inform her former husband or his father, neither was required to support the child, though this didn’t prevent it from being the father’s heir.
*The children of parents in relationships of concubinage or contubernium were considered children of unmarried parents, because, in fact their parents were not married even if they were in a recognized relationship.
**This is where I admit I know very little about how Roman law interacted with local law and custom. Anyone care to share what they know?
Paul, Opinions 2.24.1-9 - Diotima
Pomeroy, Sarah. Goddesses, Whores, Wives, and Slaves: Women in Classical Antiquity. New York: Pantheon Books, 1995.
Hunter, William Alexander. A Systematic and Historical Exposition of Roman Law in the Order of a Code. London: Sweet & Maxwell Ltd., 1803.