Dear manhhiep2509,
You ask some excellent questions, and I am happy to help.

First of all, here's a blog about what you can drop in parallel:
https://magoosh.com/gmat/2013/dropping-c ... -the-gmat/

In the sentence
"My mom bought three pairs of shoes, as many as were bought by my dad"
You asked: "As OG says the subject "people" can be understood, the subject "shoes" can be understood, isn't it. So, how do we know the subject of "were bought by my dad" is "shoes" while we do not have any parallel structure?"
That sentence as parallel structure. The idiom "as many as" (or, more generally, "as" [adjective]/[adverb] "as") creates two items in parallel. There is parallelism in that sentence, which is why we can drop the implied objects.

are the sentences correct?
(1) "My mom bought three pairs of shoes, as many as those bought by my dad"
(2) "My mom bought three pairs of shoes, as many as those by my dad"
(3) "My mom bought three pairs of shoes, as many as bought by my dad"
(4) "My mom bought three pairs of shoes, as many as by my dad"
Those are all grammatically correct. There are no grammar errors, but they are not necessarily rhetorically acceptable. First of all, the first three are way more wordy than they have to be, because the fourth is perfectly fine. I think it would be far more direct, powerful, and concise in this form:
(5) "My mom bought three pairs of shoes, as many as my dad did."
Active language always makes for a more direct and powerful structure. See:
https://magoosh.com/gmat/2012/active-vs- ... -the-gmat/

Are these sentence correct?
(1) Yesterday, the family had a picnic with more food and water than it had had for previous picnics.
(2) Yesterday, the family had a picnic with more food and water than those it had had for previous picnics.
The first is perfectly correct: perfect parallel structure. The second is awkward, primarily because of the pronoun ambiguity -- the plural pronoun could refer to "picnics" or to "food and water". Even if we dropped the words "and water", so that we only had one plural noun, version #2 still would be awkward because it adds an extra unnecessary word. Extra unnecessary words are not necessarily grammatically incorrect, but rhetorically they are abysmal, and the GMAT does not tolerate them.

Does all this make sense?
Mike