The above two pictures reveal two fathers' different teaching methods.

Apostrophe goes at the end to indicate possession.

OR you could say "two fathers AND their different teaching methods".

In the first picture, the fa-ther is urging his son to study hard, while he is smoking and watching TV idly.

In the second picture, the father and his son are both concentrating on the studying.

Studying as a verb, which they are both doing.

Apparently, the author of the cartoon focuses on athe fact that many parents whenwhile educating their children, many parents tend to neglect the impact of their own action upon their childrens.

Fact is definitive, we know what the author is communicating. Therefore we use 'the' as the preposition because it is definitive.

Keep the subject (parents) and their action (neglecting their own actions) together.

OR
"...many parents tend to neglect the impact of their own actions while education their children"

It is wWithout any doubt that, all parents hope thawant their children couldto have a bright future.

Just making it a bit more concise.

Therefore, they tend to count on schools and the society to provide their kids with a good education.

However, what they don't realize is that parents are their children's first teachers of children.

Active vs. passive voice. NOT ESSENTIAL, just sounds slightly better and more natural in this case. Typically you want to use active voice when you can, unless the actor (subject) is unknown or unimportant, OR when you want to focus on the action/recipient of the action rather than the performer. But here they are equally important.

Active voice: I eat the apple (subject, verb, object)
Passive voice: The apple was eaten by me. (object, subject, verb)

Parents who failing to set a role modelpositive examples for their children will only result in an unhealthy family atmosphere, which is definitely harmful to the future develop-ment of their children.

You could also say "be positive role models", or "set positive examples".

You don't SET a role model, you ARE a role model.

From my perspective, education from parents is of vitally importancet to a child's healthy growth.

Again active vs passive voice.

Therefore, parents should provide their children with a favorable growing environment by being a posi-tive role models.

Plural because BOTH parents should be positive role models.

Only through persistent elfforts and proper guidance can children enjoy a bright future.

The above two pictures reveal two fathers' different teaching methods.

If this were one father showing his teaching method: A father's teaching method.
But as a plural, you add an apostrophe after the s: Two fathers' teaching methods.

In the first picture, the fa-ther is urging his son to study hard , meanwhile he is smoking and watching TV idly.

I would separate these with "meanwhile" to make it clear that the second half is about the father's actions.

In the second picture, the father and his son are both concentrating on the studying.

Apparently, the author of the cartoon focuses on athe fact that many parents, when educating their children, many parents tend to neglect the impact of their own action upon their childrens can have.

I understand what you meant, but the overall sentence is a bit clunky. For a smoother sentence, stick the subject you are talking about (many parents) with the action (tend to neglect).

It is without any doubt that all parents hope that their children couldan have a bright future.

The previous sentence isn't wrong, but this sounds a bit better.

Therefore, they tend to count on schools and the society to provide their kids with a good education.

Parents failing to setbe a role model for their children will only result in an unhealthy family atmosphere, which is definitely harmful to the future develop-ment of their children.

Therefore, parents should provide their children with a favorable growing environment by being a posi-tive role model.

Only through persistent elfforts and proper guidance can children enjoy a bright future.