I started on the first paragraph... I've given you two versions. The first is yours simply corrected. The second has been enhanced for clarity.

Raising children has always been considered   the duty of women. That is why the idea that women are better parentS bearS (the word "idea" is the subject so bear must be bears)in people’s mindS. It is actually true everywhere in the world. In the last few decades, things HAVE tendED to improve over time thanks to modernization of THE couples’ ways of life, but things seem to remains the same: women are better parentS than men. (For proper writing in English, you should never start a sentence with "but" or "and", yet it is used that way everyday in conversation).

Raising children has historically been considered the duty of women. Therefore, the idea that women are better parents is still quite prevalent. It is true throughout the world. In the last few decades, there has been a modernation of the couple dynamic, yet the idea still pervades: women are better parents than men.

Women are better parents than men? IESTL essay practice

Raising children has always been considered as the duty of women. That is why the idea that women are better parents exists in people’s minds, and it is actually true everywhere in the world. In the last few decades, things have improved over time thanks to the modernization of couples’ ways of life. But things seem to remain the same: Women are better parents than men.

Basically, a good parent must take care of his or her children constantly and responsibly. To do so, he or she should provide them with all the required material things, like: a place to live, food, hygiene, health etc. Besides that, a good parent has to provide an appropriate education.
Even if the definition of the best education is quite controversial, a good parent should try to teach his or her children how to adapt themselves to the social world.
Finally a good parent needs to understand them and adapt his or her education accordingly. From my point of view, it is the key point to being a good parent.

But women and men have unequal roles toward the children's education. As women are in general more sensitive, they tend to be more comprehensive [editor's note: I am not sure what you mean by "comprehensive."]. That allows them to further adapt their behaviour to the personality of their gender. Furthermore women are more implicated [editor's note: I am not sure what you mean by "implicated."] for many reasons; for example they have this special link that was developed during their pregnancy; there is also the social influence, which constantly challenges or blames them.

However, in the last few years, men have become more and more implicated [?] in the children’s education. This phenomenon is due to the modernization of the society, which leads couples to share a lot of household tasks, but even so women remain better parents.

It is quietly common to think that women are better parents than men. It is most often the case because women are more sensitive and comprehensive [?]. This makes them more aware of children's needs. Thanks to the modernization of couples, men tend to improve themselves but women still remain better in that field.

[editor's note: Maintiens le travail dur! Vous faites très bien :) ]