Thanks to to Ngo Duy Quang for the comments. Well-spotted.

Helloworld, the essay is fine, really, with not many errors, but it could be more interesting. Here are some points first before I explain what I mean...

Word Choice/Word Form/Usage

• that teachers indeed do an indispensable role → that teachers indeed play an indispensable role OR    that teachers do indeed play an indispensable role  OR    that teachers do indeed have an indispensable role
   
• how teachers interpret those information → how teachers interpret that information
   
• young students often make conflicts with others. → young students often end up in conflict with others    OR  young students often quarrel with others

Cohesion and Fragments

Cohesion is one of the four criteria in IELTS Task 2 Writing. It basically means how your ideas and sentences join together. Have a look at the official descriptors for IELTS Task 2 Writing (public version) here, under Cohesion

 You wrote:

• And also, young students often make conflicts with others. It is always teachers that make the final decision on who is right and who is wrong.

I would join these two sentences to make the link clearer, or I would add some clearer link

Young children often quarrel at school, and it is always the teachers that have to decide who is right or wrong.

You wrote:

In terms of the amount of time young students spend with their parents only.

This is a fragment. It needs a subject or a main verb, or it needs to be joined to the next or previous sentence. Read more about Fragments or Comma Splices here.

Here’s one possible rewrite:

 Looking at the time children spend at home, it’s clear that parents are vitally important to their development.   OR
With the sheer amount of time children spend at home, it's clear that parents are crucial to their development.

Tense

I might change this sentence to make the tense more consistent:

• Teachers, in some people’s opinion, would play a more important role in the children’s development of various aspects than parents do.→

  Teachers, in some people’s opinion, play a more important role in the children’s development of various aspects than parents do.   OR     
  Teachers, in some people’s opinion, would play a more important role in the children’s development of various aspects than parents would.

Shorten/Simplify

You wrote:

I argue that parents are still of equivalent significance compared with teachers for children who start primary school. 

It's not wrong, but it doesn't flow very well. Here’s one possible rewrite to make it a bit lighter (few nominalizations) and perhaps less mathematical:

I believe parents are still as important as teachers for children at the beginning of primary school. 

You wrote this long (26-word) sentence:

Nevertheless, we should also notice that in addition to the time students spend at school, they would spend remaining parts of every day with their parents.

Again, it's not wrong, but it’s a fairly obvious point -  does it need 26 words?. Here are some possible rewrite:

Children, however, spend most of their time at home, not at school.   (12 words)  OR   
Obviously, however, children spend more time at home than at school.  (11 words)   OR
However, children spend only a small part of their day at school, and most of their learning takes place at home. (21 words)

Overall your essay length is fine at 273 words. The sentence length could be shorter – it’s 17 words on average. Add some short sentences to reduce this figure -  I recommend between 12-15 words on average.  

(You can check average sentence length, word count, and many other useful statistics by using one of the two readability links at the top of this page-  this one at read-able.com, or this one at online-utility.org.)

The tone could be lighter – it’s not a very serious topic! Try to have more verbs, fewer nouns, and shorter sentences. Remember too that the question asks you to give examples from your experience -  it might be interesting for the reader to have one or two examples from your life or family.  Imagine if you an examiner and you have to read 20 or 40 essays on this topic!

You have very few errors and your structures are generally good. Just relax and I hope we see some more essays from you here.

Thanks to Ngo Duy Quang for the comments – I agree with all the suggestions about pronouns (decide on a child OR children, but stick to one), linking ((avoid starting sentences with ‘and’), and plurals/agreement (these information).

This is a great essay in my opinion. Your ideas and academic style is good. However, some mistakes should be improved and your essay would be a great one.

As soon as a child starts(children start) primary school, they will no longer be influenced by their parents only. Teachers, in some people’s opinion, would play a more important role in the children’s development of various aspects than parents do. However, I argue that parents are still of equivalent significance compared with teachers for children who start primary school.

To begin with, I would point out that teachers indeed do an indispensable role in educating the children. In most cases, teachers are the major source for young students to acquire knowledge and information. Thus, how their teachers teach them and how teachers interpret those(that) information would affect students’ intellectual development and the way they think about the outside world. And also(In addition), young students often make conflicts with others. It is always teachers that make the final decision on who is right and who is wrong. The pattern in which teachers deal with those conflicts may even be applied by students in the future.

Nevertheless, we should also notice that in addition to the time students spend at school, they would spend remaining parts of every day with their parents. In terms of the amount of time, young students spend with their parents only. We cannot deny the importance of parents to the development of children. Most importantly, family will make children happier in childhood and more optimistic in the future. Therefore, it is undeniable that parents are as important as teachers for children development(children's development).

To sum up, it is not wise to lay more emphasis on schooling than on parenting. We should pay equal attention to both in order to help children develop well.

Btw, your topic has some relationship with mine, so you can read my essay and give me some comments, too. http://staging5.writefix.com/?page_id=2722/about-this-forum/good-parenting-has-many-positive-effects

As soon as a child starts primary school, they will no longer be influenced by their parents only. Teachers, in some people’s opinion, would play a more important role in the children’s development of various aspects than parents do. However, I argue that parents are still of equivalent significance compared with teachers for children who start primary school.

To begin with, I would point out that teachers indeed do an indispensable role in educating the children. In most cases, teachers are the major source for young students to acquire knowledge and information. Thus, how their teachers teach them and how teachers interpret those information would affect students’ intellectual development and the way they think about the outside world. And also, young students often make conflicts with others. It is always teachers that make the final decision on who is right and who is wrong. The pattern in which teachers deal with those conflicts may even be applied by students in the future.

Nevertheless, we should also notice that in addition to the time students spend at school, they would spend remaining parts of every day with their parents. In terms of the amount of time young students spend with their parents only. We cannot deny the importance of parents to the development of children. Most importantly, family will make children happier in childhood and more optimistic in the future. Therefore, it is undeniable that parents are as important as teachers for children development.

To sum up, it is not wise to lay more emphasis on schooling than on parenting. We should pay equal attention to both in order to help children develop well.