Children so have rights. Some of them are restricted. But even then you wouldn’t throw a child into prison for a crime committed by their parents.
Exactly, therefore the claim in OP that "They can't claim a fucking fetus is a person and then not give it rights." isn't really sound. Our legal systems already claim that some beings (children in this example) are persons but that they aren't entitled to the full range of rights that adults enjoy.
Please do note that I'm not trying to argue any related issues (such as the legality of abortions), just that claim made by OP.
(The systems for that are in terrible shape too, but that's a separate issue.)