In some countries there are laws prohibiting publishing a person's picture without that person's consent. The US has no such general rule, and many other countries do not either. The details of such laws, where they exist, vary from country to country.
In the US there are two kinds of legal action that might be used by a person to stop that person's image from being published. These are a suit for invasion of privacy, and a suit to enforce a person's right of publicity, also called a right of personality. These torts vary from state to state, and are not recognized at all in some states.
Invasion of Privacy
This has several variants, such as "intrusion upon seclusion", and "disclosure of private facts".
Intrusion generally applies when someone has entered a private place, such as a dwelling, without permission, and has then taken a photograph, made an audio or video recording, or perhaps a written account, and has published or attempted to publish the image, recording or account. If and only if the publication would be "highly offensive to a reasonable person" then the subject or lawful occupant of the place invaded may be able to obtain money damages, or an injunction against publication. This would not apply to the fact pattern in the question, because the parents would have had a right to be in their own home, or anywhere that such baby photos are likely to have been taken.
Intrusion can also apply to one who observes private activities without permission, even if no recording is made. But the "highly offensive" standard still applies.
A "private facts" case generally applies when a person (or a small group, such as a family) has attempted to keep certain facts private, but some other person or business has published them (or attempted to publish them). A successful suit will in most states that allow such suits require evidence of some positive effort to keep the facts private, and that the publication would be "highly offensive to a reasonable person". This might possibly apply in the fact pattern described in the question.
But note that the "highly offensive" standard applied to both of these privacy torts. Most typical baby pictures would not be considered "highly offensive", this is a fairly high bar. It would be a decision ultimately made by the jury or other finder of fact, I believe. For comparison, peeping on a couple having sex in a hotel room has been found to be highly offensive. When a person had sex with another, and had secretly arranged to record a video of the sexual encounter, and later distributed it, that was found to be highly offensive.
Right of Publicity
This generally prohibits using a person's, name, image or likeness, or "persona" to endorse or advertise anything, or for other commercial purposes, without consent. It is most often applied to unauthorized celebrity endorsements. Most US states have some form of this, but the details vary significantly. Unless the baby pictures were being used to advertise something, or in some commercial manner, this would not apply. If the parents sold baby pictures to someone who used them to advertise, say a day-care, or some baby product, this might apply, although it might be that the parents would be held to have had the right to act as the child's agent. If someone else downloaded the baby pictures from Facebook (or any other such site) and used them in ads, this might well apply.
It is unlikely in the US that a child could successfully take legal action against a parent for having posted pictures of the child, even if they were embarrassing, unless they were also "highly offensive". Even then such a suit would be unusual, and might not be successful.
It is more likely that a request to the parent would be successful, unless the parties are on vary bad terms. A request to the site to take down such images might work, depending on the site's terms. A DMCA takedown would not work, because the subject does not normally hold copyright to an image, the photographer initially does (unless it is a work made for hire, in which case the employer initially does).
This conclusion is specific to US law. The outcome would be different in at least some other countries.