In English, an auxiliary verb like do lets us omit the rest of a verb phrase:
Do you like ice cream?
Yes, I like ice cream.
or
Do you like ice cream?
Yes, I do like ice cream.
Adding the auxiliary verb do lets us omit the rest. This is called Post Auxiliary Ellipsis.
In my example, you could also say:
Yes, I do like ice cream.
That is, you can add do without omitting the rest. The ellipsis is optional.
Do is appropriate without omission in this example because of the emphasis on whether the clause is true or not. This is often the case in response to a yes-no question. But in your example, things are a little different:
The Moon illusion is an optical illusion which causes the Moon to appear larger near the horizon than it appears higher up in the sky.
The Moon illusion is an optical illusion which causes the Moon to appear larger near the horizon than it does appear higher up in the sky.
We can replace appear with does and it sounds fine. But if we add does without omitting appear, it sounds weird:
？The Moon illusion is an optical illusion which causes the Moon to appear larger near the horizon than it does appear higher up in the sky.
I've marked this example with a ？ symbol to indicate that it's questionable. In this case, adding emphasis on whether the clause is true or not is very strange―we're not answering a yes-no question, we're in the middle of a comparative construction ("X-er than" / "more X than"). Adding does here makes the sentence strange and hard to understand.
By the way, remember that I called this Post Auxiliary Ellipsis, and not Post-do Ellipsis? That's because any auxiliary works, not just do:
Are you going to the store?
Yes, I am going to the store.
In this example, we already have the auxiliary be, so we use that one instead of adding do.
Of course, we could come up with another theory to explain the same facts. What if we wanted to keep things as simple as possible?
The Moon illusion is an optical illusion which causes the Moon to appear larger near the horizon than it appears higher up in the sky.
The Moon illusion is an optical illusion which causes the Moon to appear larger near the horizon than it does higher up in the sky.
We could just say does replaces appears. If we do that, we're calling do a pro-verb. A pro-verb is like a pronoun, except it substitutes for verbs (or verb phrases) instead of nouns. And in English, do is our pro-verb of choice.
This explanation works pretty well, as long as we're willing to admit other auxiliaries as pro-verbs:
Are you going to the store?
Yes, I am.
Now auxiliary be is a pro-verb substituting for the entire verb phrase be going to the store, at least according to the pro-verb theory.
Best Answer
Let's discuss the differences between several possible alternatives, starting with:
That's a bit awkward for what you're trying to say. Consider an example with the opposite - "John's mother was overjoyed to see him alive after coming back from war." What you probably want to say here is "imagined the sight of them dead", or "had a vision of them being dead". Consider also this example: "Steve's mother would always worry whenever he jumped on his motorcycle. No matter how hard she tried to repel bad thoughts, she always saw him dead in a ditch somewhere." Given that John's parents have already been killed and he knows it, the situation is no longer hypothetical like in that example, which is why the sentence sounds odd to me.
That's talking about a witnessed event in the past, and would imply that John actually saw his dead parents in real life, and didn't just imagine the sight of it.