Generally speaking, the more upscale the park/campground/resort, the more likely they are to have limitations and restrictions on what sort of RV you can use. A schoolie is likely to be welcomed at state and national parks without a second thought, assuming it's in anything resembling halfway decent condition.
The RVIA is a trade association, and the RVIA seal mostly means that the maker is a member of the association and hence paying dues, etc. It also is a sort of self-certification that they are meeting legal requirements in terms of fire, electric, sanitation, and motor vehicle codes, with some (I suspect very minimal) verification/oversight by the RVIA.
I wonder if parks that require an RVIA seal (specifically) frequently turn away those commercial RVs made by companies that are not RVIA members and hence do not have the seal? Some of them are perfectly decent makers.