Is it legal for a parent to take off a 15 year old's bathroom and bedroom doors and place a camera that looks into the bathroom and bedroom?
The relevant law is not so specific. It prohibits child abuse and child neglect which are defined only as general standards and not as specific rules.
This doesn't appear to be child neglect, indeed, the opposite to the extent that there is such a thing.
So, would it be child abuse? This would be up to the finder of fact to determine, and might depend upon the manner in which this is done (for example, what is said to a child about it) and the reason that it is done (e.g. a history of self-harm) and more generally in light of the total context of the situation.
But it is not obviously child abuse, unless, for example, recorded video was used for child pornography purposes, which there is nothing in the question to indicate.
A comment suggests that the criminal offense of voyeurism (F.S. § 810.140) or video voyeurism (F.S. § 810.145) might be implicated, but both of those statutes apply to "secretly" observing someone or "secretly" setting up cameras, while in this case, the cameras and viewing are anything but secret. So, even if it were child pornography, it would not be voyeurism or video voyeurism under state law in Florida.
As a general rule, a child is not entitled to privacy from a parent except in certain specifically defined circumstances (e.g. certain privileged communications).