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Education isn’t deciding which authorities to trust — that is literally authoritarianism. The point of education is to make your own good decisions in unclear contexts.

I agree in principle, but most people do not have the necessary education to make accurate judgments across all fields from first principle reasoning. Eg someone with a PhD in physics probably doesn’t have the knowledge to debunk every argument against the COVID vaccine. In those cases we naturally rely on asking some biologist who does have that knowledge what they think.

I have a similar point to make as the other comment. Not everybody has the resources to e.g. verify the germ theory of disease. In fact humanity didn't have the resources to verify that until recent history. The smartest, brightest, and most successful medical practicioners would use to tell you that disease is caused by an imbalance of the humors, or of yin and yang; this was a comparatively obvious and reasonable, but ultimately wrong explanation supported by anatomical evidence, compared to the fanciful explanation of invisible insects infiltrating and attacking your body; a theory that requires extensive institutional knowledge, trust and infrastructure to develop and verify.

So when a government and its politician leaders frequently lie to its constituents, it can be difficult for a rational individual to trust the public health advice coming from the same apparatus.

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?

How are they "clearly recognizing" these "experts" as "untrustworthy" without the expertise to recognize which so-called experts are bogus and which ones aren't?

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?

If public health experts are caught being deceptive in one case, or the media publishes as truth something that ends up being political propaganda- even once - they lose trust and people will correctly choose to treat future information from that source with skepticism.

   even once

This isn't realistic in the slightest, and is the diametric opposite of "correct." What information source is one hundred percent accurate?

By this standard, people can't trust literally any source at all. This is an absolutely sick and objectively poor way of thinking and wind up just sort of believing whatever they want to believe. Whatever feels good, or, is the least bad. Or feels bad, but gives them a sort of predictable sort of bad feeling -- rage and an ill defined sort of aggreivement about some sort of ill defined state of affairs that has been denied to them by ill defined people for ill defined reasons -- that is at least preferable to the work of actually knowing things and understanding how anything works.

So are they Fox News people, or have they moved on to the harder stuff?

They are less intelligent, regardless of the country. This anti-establisment-ism is based on a fundamental inability of making logical steps longer than 2, or the inability to generalize. Typical wacko seem one single example of something suspicious, maybe even a fact, then a scammer wraps that suspicious event or a fact in a conspiracy theory, and that person believes it instantly, not questioning this simple and crude world view.

Two most popular examples nowadays:

1. Moon landing. A person sees 1 single moon picture with unexplained details on it, read a conspiracy explanation about it, and then makes a single logical step that a single picture falsifies everything about moon landing. It doesn't enter his mind to think about the other 9999 already explained photos from the Moon, and then to ask that conspiracy guy how do those fit in his theory. It doesn't enter his mind to maybe question his skills at deciphering optical illusions and artifacts and ask for help.

2. Vaccines. Antivax crowd prides themselves in finding some negative cases for any given vaccine, be they fake or real. But even in the case of real deaths for example, they intellectually unable to take a second step and apply statistics, to understand how much more good a vaccine brings to the humans compared to the bad.

They can't really reason logically, aside from simple "single fact" -> "decision". Anything more complex is no go.