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I'm reading When Our Grown Kids Disappoint Us by Jane Adams. In the introduction, she talks about who this book is addressed to:

...here are the positive, life-affirming, burden-easing things you can
learn from the experiences of hundreds of parents of kids who've let
them down. That sounds like a self-centered way to describe those
parents and the choices their kids have made, doesn't? After all,
whose life is it, and who are we to judge how they should live it?
Just because they didn't finish school, or marry the right person ...
who are we to say they've failed? Only their parents, for whom
coming to terms with our adult children's limitations also means
facing our own. Although their names and the details of their lives
have been changed, they are as real as you and I, with real kids and
real disappointments...

I'm confused about the part of "Only their parents". Is it the answer to the question "who are we to judge"? So the author is saying that only the parents of those kids can judge them? But it also seems the author is implying that parents do not have the rights to judge their kids. Can anybody explain the idea for me?

Breaking down the relevant part of the passage:

whose life is it, and who are we to judge how they should live it? (...) who are we to say they've failed? Only their parents

Here the question is clearly

whose life is it, and who are we to judge how they should live it? (...) who are we to say they've failed?

However this question is rhetorical, because no answer is expected here and what follows is not really an answer to this question. With this rhetorical question the author actually means to say that

it is their life: we cannot judge them or say that they have failed.

That logically leads to another question which for some reason is not actually stated by the author in this passage, probably because it is implied by what comes before:

Then who can judge them// who can say they have failed?

And the answer is

Only their parents (can do that.)

Here the actual unstated question is implied by the rhetorical questions that come before, and the answer that follows.

So your reading is correct:

the author is saying that only the parents of those kids can judge them.

Coming to the final part of your question,

But it also seems the author is implying that parents do not have the rights to judge their kids. Can somebody explain

In fact the author is saying that the parents do have the right to judge their children (which is a right that nobody else has, as noted earlier) but the parents themselves suffer the consequences of doing so, because

coming to terms with our adult children's limitations also means facing our own (limitations as parents)

as rightly pointed out by @peterflynn in the earlier answer.

As far as I can tell, yes: "only their parents" is answering the question "who are we to judge" / "who are we to say they've failed?"

I think the implication is meant to be that, in judging their children's "failures," really the parents are coming to terms with their own failures. When parents feel like their children have failed or disappointed in some way, that is inseparable from the feeling that they have failed their children by falling short as parents as well.

At least, that is what I take it to mean – it's definitely left a bit unclear. I would hope the author explains her viewport further in the following paragraphs...