How could there be meaningful parallels between quantum mechanics and eastern philosophy/religion?

[Long text ahoy]

There are people who say some stuff in QM sounds like some stuff in eastern religion, like the Vedic texts or Daoism. Some of them are very smart and respectable. Books and articles have been written. Never read any of them, no idea if they're right.

Question is, how could such parallels, if they exist, be meaningful? I assume the idea behind pointing them out is (sometimes? often? always?) to suggest that ancient wisdom presages modern knowledge, going into a defense of religion, spirituality, mysticism, etc. That's cool and all.

But how could ancient contemplative practices have afforded the same knowledge as 20th century physics? I'm gonna be totally naive here: didn't those physicists use big machines and high tech equipment and a complicated mathematical conceptual apparatus (or, like, whatever, man) to make their discoveries?

Say you're a quantum field theorist who loves his Daodejing and wants to show how ole Laozi really had it all right way back when and it's not so weird to be a new timey scientist as well as an old timey spiritualist. What are your philosophical options for accounting for how there could be this convergence or whatever it is? How could Laozi have made the same (similar?) discoveries with the (presumably) radically different methods available in his time?

If you're not gonna postulate some kind of literal "second sight" or 6th sense type phenomenon where meditation or mystical experience just gives you straight telepathic knowledge or divine revelation or something -- if you're not gonna be extravagant or supernatural with your metaphysics -- how do you handle it? Do you go transcendental idealist and say since nature is in my experience, meditating deeply on my consciousness is really the same as studying the basic constituents of spacetime/matterenergy/lawsofphysics? Or what? What are your actual possibilities here? (Or is it presumptuous of me to think these physicists aren't gonna want to be "extravagant or supernatural with their metaphysics"?)

PS. I'm a phil undergrad so answering with some jargon is OK but bear in mind I know almost nothing about the topics I'm asking about (what QM says, what ancient eastern religious texts say, what the alleged parallels are, etc).