The argument was put forth in this article by @kyriacos that abortion is a natural right based on several perspectives.

Source drsuparna http://www.flickr.com/photos/74896762@N00

The unborn is a parasite
The right of choice/freedom - "Nobody gives you the right to decide for the life of others."
His conclusion:

Abortion is an issue of entitlement. Some people believe that they can have a say about how other people treat their own bodies. The drama behind it is mostly for religious reasons. Most believers feel the right to “guide” other people’s lives because they feel that we are part of a greater plan. Truth or ethics are not and cannot be democratic if we value individuality. No matter what one believes about the value of life, no one has the right to impose their will on others. Our bodies are our only sacred property—not the will of others.

Most of his argumentation is found in the comments that followed, so I won't list them here. I will, however, address some of them in this counter-argument, which finds his argument flawed on every level.

Science

The claim that the unborn child is a parasite is actually only partially correct. It may be argued that there is a parasitic relationship, however, it is clearly not wholly so. A simple Google search provides tons of scientific material for this discussion. An honest assessment of the vast majority of it clearly makes the case that the unborn is not a parasite. Search results here.
First, the child is not invasive. It did not originate elsewhere. It's life was the product of the mother. Parasites are not dependent upon the host for their origin. If one attempts the "tumor" argument, then it must also be pointed out that tumors are unhealthy spontaneous results, contrary to pregnancy. Furthermore, tumors are part of the host's body, sharing the same DNA.
Second, the relationship is not purely parasitic. It is, in many ways, symbiotic. This is because the placenta regulates the nutrients between the mother and the child, striving for a balance that ensures the health of both. The function of the placenta will actually favor the mother's health over the child's. A study of the p110a gene will help bring more light to this, for those who care to dig further. Cambridge University did an extensive study on it.
Third, cells under a fingernail have the exact same DNA of the host from which they were "scratched". The unborn has its own unique DNA, distinct and yet wholly dependent upon the mother's. This means it is a distinct life, while at the same time distinguishing it from a true parasite.

Human Rights/Freedom

This is pretty basic. Either murder is objective or it's subjective. If murder is the taking of a human life, then abortion is murder. Otherwise, it must be given subjective parameters.

Pixabay

Libertarians for Life, a non-religious group, offers a sound argument against the parasite argument from a classic liberal perspective here. There's plenty more to read, here.

One popular misconception is that libertarianism as a political principle supports choice on abortion. And major elements within the libertarian movement (the Libertarian Party, for example) take abortion-choice stands. Nonetheless, libertarianism's basic principle is that each of us has the obligation not to aggress against (violate the rights of) anyone else -- for any reason (personal, social, or political), however worthy. That is a clearly pro-life principle. Recognizing that, and seeing the abortion-choice drift within the libertarian movement, Libertarians for Life was founded in 1976 to show why abortion is a wrong under justice, not a right.

Dr. John Cobin, Phd (Public Policy & Economics), provides more insights from a libertarian public policy perspective in his excellent book, Pro Live Policy. You can view/download it here.

Abortion is the taking of innocent life on purely philosophical grounds as part of the doctrine of natural rights. Hence, the issue of abortion does not fall in the pale of religious sentiment, but has to do with the right innocent people have to live. If a pluralistic society has any common goals, the preservation and protection of innocent life must be one of them.

EDIT: When I informed Dr. Cobin that I had used this particular material, he suggested I add his article, Abortion Policy and the Market, published in the Journal of Interdisciplinary Studies. He thought it would be more "credible" for readers.

The only truly scarce resource is the human mind. All other things that are commonly considered to be scarce can be found in more abundance, synthesized, or replaced through the powers of the human mind (Simon 1996). Yet abortion is perhaps the most potent enemy of the human mind, as abortion policy destroys the one thing in life that truly cannot be replaced.

If we affirm the non-aggression principle, we cannot aggress against the life of another. Contrary to the "parasite" argument, the child is not aggressive. It is dependent, for sure. It shares nutrients, without doubt. But it does so, rather than invasively, passively as a good and necessary product of procreation (the actual free "choice" in this discussion).

It is tragically fascinating that someone can argue repeatedly that nobody has the right to control the mother's body while at the same time claiming that the mother has the right to control another's body - one that is solely the result of her activity and dependent upon her because of her. What a horrific irony.

Legalities & Modern Medicine

This one is actually pretty easy, in spite of Roe v Wade. A simple list of medical reasons an unborn child is a human being, from here:

Freeimages

Unique DNA
Organs distinguished by three weeks
Heartbeat in 18 days
Facial features at four weeks
Circulatory system developing at five weeks
React to touch at six weeks
Brainwaves deteced at six weeks
Ear observed at six weeks
Movement clearly evident at eight weeks
Yawn and suck thumbs at nine weeks
Fingerprints at ten weeks
Pulls away from invasive abortion procedures at thirteen weeks

Biologically speaking, the life of a human being begins at the moment of conception in the mother’s womb, and as a general rule of construction in the law, a legal personality is imputed to an unborn child for all purposes which would be beneficial to the infant after its birth.... A child unborn at the time of the death of its parent has also been considered a “child” of the decedent in determining beneficiaries of an award in a wrongful death action or in a workman’s compensation case.
^{42 Am. Jur. 2d, “Infants,” sec. 2. from here, as are the two following quotes.}

Medical authority has long recognized that an unborn child is in existence from the moment of conception.... All writers who have discussed the problem have joined in condemning the total no-duty rule and agree that the unborn child in the path of an automobile is as much a person in the street as its mother, and should be equally protected under the law.... Most courts have allowed recovery, even though the injury occurred during the early weeks of pregnancy, when the child was neither viable nor quick. Viability, of course, does not affect the question of the legal existence of the unborn, and therefore of the defendant’s duty, and it is a most unsatisfactory criterion, since it is a relative matter, depending on the health of the mother and child and many other matters in addition to the state of development.
^{Prosser and Keaton on Torts, 2 ed., sec. 36 (1955).}

If the mother can die and the fetus live, or the fetus die and the mother live, how can it be said that there is only one life? ...The phenomenon of birth is not the beginning of life; it is merely a change in the form of life. The principal feature of that change is the fact of respiration.... A baby fully born and conceded by all to be “alive” is no more able to survive unaided than the infant en ventre sa mere. In fact, the babe in arms is less self-sufficient–more dependent–than his unborn counterpart.... The fact of life is not to be denied. Neither is the wisdom of the public policy which regards unborn persons as being entitled to the protection of law.
^{O’Neill v. Morse, 188 N.W.2d 785 (Mich., 1971).}

Linked above, Constitutional Personhood of the Unborn Child by Robert C. Cetrulo, points out that flawed legal and medical backing of abortion by providing evidence of human life and laws to defend it. Of course, we don't agree with all laws and may (and may have a duty to) disobey them if they are immoral. But the argument should carry a great deal of weight, regardless of where one finds themselves in the discussion.

Princeton has a page dedicated to the affirmation that "human life comes into existence with the formation of the one-celled zygote".

Conclusion

I could provide more, but few will even bother to read what I've submitted to this point. The argument for the "right" to abort is flawed on multiple levels. It's an emotional plea for freedom couched in the rights of one person, with no regard of the destruction of another that exists solely as a result of the first person's actions. In essence, it's the wanton destruction of life and freedom in the name of life and freedom.

Reflection

Future generations will look upon this period of time as one of the great atrocities of history. Like slavery, the many genocidal efforts and war-mongering for profits, abortion in the contemporary westernized cultures, the snuffing out of the lives of more unborn per day than died on 9/11, will instill a sense of wondrous horror on historians who study our times, incredulous that a people making such great advances in so many spheres of science could fail so utterly tragically and grievously in this one - based purely on the scientific evidence that has stared us in the face for many decades.

Steemin' on,
Another Joe