I’m agnostic on the question of whether “we can always tell”. My wife, a high school teacher, was pretty sure she could “always tell” with respect to plagiarism, but then that was mainly because she knew the kid from the classrooom. A reader of essays in a college app would not have that knowledge.
Is it therefore possible that such a reader could inaccurately pin this label on a kid? Possibly. But that’s really beside the point, as I see it. I would suppose they’re looking for much more than grammatical correctness and stylistic perfection. That’s what Nondorf was signalling when he referred to “not sounding like a 40-something lawyer”. It was not about detecting cheating so much as about what makes for effective essays from their point of view - youthful exuberance, I would guess, intellectual curiosity, certainly - spirit, inchoate goals, something of the kid’s unfinished view of the world and self. That famous independent-mindedness, quirkiness. Perhaps there are a few forty-somethings out there with Shakespearian imaginations capable of mimicking a youthful take on these things in a convincing and appealing way - but not many. A parentally authored piece would probably sound a bit phoney, a bit dead and unconvincing, and that would not have much to do with grammar and word-choice. It would not be appealing in its own terms, even if it did not damn the kid as a cheater. I take Nondorf as saying that your parent cannot convincingly express what you’re about; only you yourself can do that. Therefore keep your parents out of it, and put yourself in it!
“We can always tell” may not be literally true, but it sounds to me like a useful warning all the same, both on ethical and functional grounds.
Edit: I agree with @CU123 at #9, said with less verbosity.