It's a typically religious argument, but even if you drop the religious component, there is still the issue of roughly where you draw the line. It also depends on what your criteria for value are. If the criterion in question is a being's humanity, then it's conceivable to hold to hard line anti-abortion positions. If you think something's humanity is not a strong enough criterion, you might use the more philosophically sophisticated notion of "personhood."
Historically, Christianity has had problems with abortion. The question of when "ensoulment" occurs has plagued their theologians for some time. In Medieval times, some of their thinkers believed it occurred when the fetus first moved independently, and before that the fetus was just a lump of tissue. More recently, lots of Christians have come to the conclusion that it basically happens at the moment that the gametes fuse (the claim that "life begins at conception"). However, Christians don't talk about "ensoulment" as such these days, because we live in a secular society, and you can't sell openly sectarian notions here quite so easily. So they use the more neutral term "life" (even though, of course, the gametes were alive prior to the creation of the zygote, and the zygote is alive before it becomes an embryo, and so on).
The rough idea is that no individual gamete on its own can become a natural-born human being, but a zygote can. Therefore the zygote should be accorded all the basic rights and protections accorded to any other person in our society. Of course, this ignores the fact that many fertilized eggs are deposited improperly, causing things like ectopic pregnancies. It also seems to potentially commit the genetic fallacy (a zygote or an embryo is no more a person, it can be said, than a seed is a tree).
Regardless, the religious people advancing this argument are just favoring one extreme of the spectrum. But assume you want to be more rational about it, or at least have more nuanced judgment. Suppose you admit that, yes, an embryo is not a person any more than a seed is a tree, but whereas it may take a long time for a seed to become a tree, it doesn't take very long at all for an embryo to become a person. And suppose, as most people do, that you rule infanticide out from the get-go. The rapidity of the development means that it becomes somewhat important where you draw the line. How close to the line of birth are you willing to go before you say abortion is basically morally equivalent to infanticide? If you just adopt birth as the line of demarcation, then you're openly admitting to the arbitrariness of the line you choose to draw.
But there are problems with any line you might want to draw. Trimesters are equally arbitrary in their own ways, and we simply don't have the tools to determine when the fetus has a sufficiently developed neural system to, say, feel pain, or have cognition of any recognizable sort, or even necessarily to survive independently outside the womb. So we're left throwing darts in the dark. When does the mass of tissue we call a fetus have the qualities we recognize in other persons? When can it survive outside the womb?
You see the problems that this runs into. Pro-choice hard liners, by contrast, will of course object that this is in some sense irrelevant, because it's the woman's body. They'll bring up thought experiments such as the dialysis patient whose survival is dependent on being hooked up to one of your kidneys and things like that. But this too is a form of extremism, since it arbitrarily privileges one person's life, or specifically one kind of person's mode of life, above that of another.
In any event, I'm basically comfortable with drawing the line at the arbitrary third trimester, since I'm sufficiently uncomfortable with the arbitrariness of drawing the line at birth, but then again in extreme cases I would consider infanticide justifiable (although not desirable). There are arguments which I find compelling that make it largely indistinguishable from late-term abortion, and if you're okay with late-term abortions, then...
All that said, yes, I do agree with you, the attacks on Planned Parenthood are rather ignorant and unfortunate. That organization offers much-needed services to women that they often can't get as easily anywhere else.
