The beginning of your post is so insightful, and I've rarely seen it expressed so well. The anti-science Americans are not dumber, and treating them as dummies is part of the problem. (One that I'm guilty of on occasion)
This is the internet, though, so here comes the part where I disagree with something. This is where you lost me:
Why should my parents, who are quite intelligent
but never received any significant science education,
believe what they are told about science by experts
that they can clearly recognize as untrustworthy?
If you said they simply don't know who to trust, because there is a lot of conflicting information (much of it deep domain knowledge) and the government is not always trustworthy or effective, I'd agree, but it seems like you're saying something a bit different?