The pro-choice conceit
Most opponents of abortion argue that kill the fetus is wrong because the fetus is a person. Pro-choice advocates often concede the personhood of the fetus but claim that the woman has a right to remove it because it is a parasite.
Once personhood is conceded, however, the pro-choice argument loses. There can be no justification for killing someone who is dependent on you because of an action you took. Worse, there can not even be a justification for killing someone who is dependent on you because of an action done to both of you by a third party.
What I mean is that killing a person is murder. Making a person dependent on you due to your action, such as voluntarily engaging in the procreative, does not justify murdering them because of the dependence. Your choice to use birth control, none of which are 100% effective, does not justify murder.
Murder cannot be justified because of the actions of a third party. This means that even in the case of rape, if the fetus is a person, murdering it because it is dependent on you is not justified. You are both victims of the rapist.
This conceit is not even necessary. I think a very reasonable argument can be made that a fetus is not a person from conception. It is much harder to claim that personhood starts at birth when there not a significant developmental difference in the baby in the instance before both compared to the instance after.
That means there is a window of time after conception where terminating the pregnancy is not murder. I am not interested here in trying to determine when exactly personhood starts, only in refuting the claim that it starts at conception.
I have two arguments that I would like to make. These arguments address the question of personhood from two sides of the question, when does it start and when does it end. I will start with the latter first.
When does personhood end?
One important aspect of being a person is bodily anatomy. As a person, you do not have the right to stick a knife in me without my consent. But personhood ends when we die.
When a person becomes brain dead, the body is still alive. In fact, the body of a brain dead person might have a better case to be made for personhood than a newly formed zygote. However, we have no problem slicing up a brain dead body to harvest the organs. We do not try feel the need to maintain the living body but mourn the person who died.
But isn't being alive the marker of personhood. Obviously not. It is something in the brain that makes you a person. When that part of the brain is dead, you are no longer a person even if the brain stem is intact and the heart beats and the lungs breath.
When does personhood start?
The main claim is that personhood starts at conception. There is a big problem with claiming that a new, independent person begins at conception and that is twins. Identical twins occurs when a zygote separates into two independent sets of cells.
What happened to the person who used to be there? Are the twins just separate parts of the same person? Did the original person die and two new people begin to exist? The most reasonable conclusion is that they are people until they reach a specific point in their development, at least sometime after cells differentiate. Once cells begin to differentiate, the zygote cannot divide into twins.
Personhood is all in your head
It is clear that personhood is a mental faculty. We can debate when that mental component that makes you a person forms, but it is not at conception. If you disagree, you must admit that a person is still a person even when they are brain dead. You would be as obligated to oppose harvesting organs from a brain dead person as abortion. No one has taken this stance.
The answer is that all human beings are created in God's image. "Personhood" isn't really a coherent concept.