Relative Clauses; there are three components:
Component 1. It is headed by a relative pronoun ("who," "whom," "whose," "that," or "which") or a relative adverb ("when," "where," or "why").
(This links it to the noun it is modifying.)
Component 2. It has a subject and a verb.
(These are what make it a clause.)
Component 3. It tells us something about the noun.
(This is why it is a type of adjective.)
Ref Grammar Monster
Quite often, the relative pronoun can be omitted. However, with a relative clause, it is always possible to put one in.
Who "answering their parents back"
This does not seem correct.
Who "answer their parents back"
This seems to work OK
Therefore the first example is incorrect, at least in it's shortened
Lets try again
that misbehave and become rebellious, answering their parents back,
This sentence is not correct as we are flip-flopping tenses; misbehaving, rebelling, answering.
that misbehave, become rebellious, and answer their parents back
The sentence works fine
Some children (that) misbehave, become rebellious, and answer their parents back because of the wrong parenting style that spoils children.
Note I have put HAD before misbehave not who before answer. Just because you singled out the phrase "and answer their parents back" does not erase the fact that the rest of the clause still exists.
Some children (that) misbehave, become rebellious, and answer their parents back.
This sounds good but the prolonged sentence did not so we will try an alternative.
Some children (that) misbehave, become rebellious, and answer their parents back may have been spoilt by their parents.
The cause must have happened before the event so the past tense been. If you wish to expand on why they have become spoilt brats add another sentence. Trying to cram too much information into one package is not always successful.