If you're talking about the oral contraceptive or the Depo-Provero shot, then no. Any person under 18 cannot legally obtain a regular prescription medication or a regular shot (such as a vaccination) without a parent's permission. You can always have a reaction to any type of prescription drug. As for something like an IUD, I'm not sure, but it seems to me that the whole deal with putting it in and taking it out would be considered a medical procedure of some kind; if not, it still must be done by a doctor, meaning the girl must be 18 or have a parent give permission. (This probably applies to a diaphragm, which also must be fitted by a doctor, so I'm told.) I think anyone can buy a condom, but there's always the embarrassment in marching up to the counter with it (my fiancé used to work in a gas station--he saw plenty of that). If you're getting straight down to what most people think of as actual abortion (though the pill, the shot, and the IUD, and probably a few others also fit under this, being kinds of abortients themselves), an under-18 person cannot undergo any other medical procedure (like an operation--even a necessary one), get some part of their body pierced, get a tattoo, go to a tanning bed, or even get their driver's license without a parent signing for them, and they cannot legally buy cigarettes or lottery tickets at all; all of these, if handled carefully either have no consequences, or have minor ones or it happens years down the road. An abortion, which is a medical procedure, immediately affects a woman's life; it not only causes psychological damage, but also puts her life in danger--it's just straight out painful, and it can also cause severe bleeding and harm future reproduction (I'm sure there are more than that; this is what I'm thinking of off the top of my head.)
So I guess the overall answer as to whether or not a girl under the age of 18 should be able to obtain birth control without parental consent would have to be no, unless they decide to start allowing young people to do other "dangerous" things without their parents' permission.