Is this sentence correct? How to parse this sentence?

Yes, it's correct.

After a few months = adjectival phrase modifying the whole sentence

they = complete subject

added up = phrasal verb

all = simple object

(of) the times = prepositional phrase with preposition elided, modifying "all"

(that) the young people got sick = relative clause with elided relative pronoun, modifying "times"

I hope that helps.

How can "(that) the young people got sick" modifies "times"? Because I think it can't be "The young people got sick times"?

It's "the times the young people got sick."

Infra is correct. It's specifying which times.

Had gotten sick

"Had gotten" if you want prescriptive grammar

Yes, it's correct.
added up => phrasal verb