No, they never did that. Was Martha supposed to hide her pregnancy?

Yes. Canonically, the Kents discovered the baby, and then their very rural area was hit by a massive blizzard that kept them isolated for six months. So they decided to just say that Martha got pregnant and pretend that Clark was their natural-born child.

Superman's origin story is that the Kents found him, adopted him, and raised him as their own. There are no parallels to today's illegal immigrants.

Except for the fact that this makes Superman an illegal immigrant. True, very few of today's illegal immigrants are literal aliens that arrive by spacecraft from an exo-solar planet. But his story does parallel theirs in that he is a non-naturalized person who is present in the contrary in violation of our immigration laws.

BTW, my family went and saw the movie this weekend - and I think many conservatives are largely missing an opportunity in getting upset over this point, rather than fully embracing the film as advancing their preferred framing of immigration. That discussion involves spoilers, so maybe we'll return to it in a few weeks when more folks have seen it.