Theory on stars

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One test might be to look at the night sky at opposite ends of the earth. And compare the parallax effect of the planets with that of the stars. If there is a parallax effect in the planets but not the stars, this means the stars are far away. If there is a parallax effect in both this means the stars are close by. If there is no parallax effect in either then further investigation will be needed.

Another possibility is that the stars are close by but no parallax effect is observed because the stars are all roughly equally distant from the earth. But this would not conflict with the idea that the earth is orbiting the sun.

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Is it observed in the planets?

Allen wrote

there is generally no parallax effect

https://en.wikipedia.org/wiki/Stellar_parallax

Anything wrong here?

Wikipedia wrote

Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years

If stars are that far away, then any observed effect is going to be so tiny that you will need to be an expert in astronomy to be able to tell if it is "legit". I think Allen does not trust astronomers, and is only willing to fully accept results of experiments he can understand, in the same way that a vaccine skeptic might want to understand the medical studies himself rather than trust the conclusions of a medical expert.