In the current climate, in my space, it's much more the former list, with the additional, and perhaps more important, addition that it is not about a parent getting to control what their child does or does not learn at the public school or library, but instead about ensuring that the most reactionary, bigoted adult (regardless of whether they have children) has unbreakable control and the ability to dictate what all children do or do not learn at the public school or the library. Regardless of the learned, considered opinions of the professionals working in those spaces. "Parent's Rights" people are almost always "Don't Say Gay" book banners and burners angling for the power to enforce their views on everyone.