The New Year is a traditional holiday and wais supposed to be a happy occasion, but it has gradually beencome a burden for young people- - especially the 80s generation.

"was supposed to be" is okay, but "is supposed to be" is more accurate, saying that it STILL should be a happy occasion, not just in the past

Both the young couple's parents hope the young coupleIt's especially difficult for young couples, because the parents from each side of a couple hope that they can celebrate the New Year with them, but they two families don't want to celebrate with each other.

"Both the young couple's" makes it sound like you're talking about two specific young couples
My change clarifies that you mean the family of each person in the couple
This is actually really hard to word naturally lol, huh...
In the second part of the sentence, who "they" and "each other" are is unclear. Are you saying the children don't want to celebrate with the parents? That the two different families don't want to celebrate with each other? (I assumed this but idk)

This always confuses the young couple and even makes them to fight;.

I would use "can" instead because it's something that may but doesn't always happen, but you can say "always" to exaggerate.
Removed the descriptor "young" because we already know the couple is young from earlier, so it sounds repetitive

Secondly, lots of young people live in big cities andwhile their parents live in villages or small towns.

"and" is okay
"while" is better because it contrasts the two statements

Due to the rapid economyic development over the past 20 years, the quality of life in cities has surpassed that in the places where young people were born.

So they feel very uncomfortable staying where they were born;.

Thirdly, because of the traditional concept of men being valuinged over women, elderly parents are partial to their son's family.

For example , they make their daughter's family do more housework, but give their daughter-in-law more smile faces and kind words.

While staying atwith my wife's family , my wife's sister-in-law is always better than her, and their daughter is always smarter than my daughter. lol.

or "while staying at my wife's family's place"
Now, we do say stuff like "I'm staying at my sister's tonight." and you don't have to add "place/house", but for some reason saying "I'm staying at my wife's family's tonight." sounds kinda weird to me... I suppose it is grammatical though...

I wish our elders' conceptsway of thinking would change one day!

The concept itself will still be there, unchanged. It's the "way of thinking" or "what they believe in" that will change. Aka, they'll stop believing in the concept.

Chinese New Year is a traditional holiday and was supposed to be a happy occasion, but it has gradually beencome a burden for young people -especially the 80s generation.

I feel it helps readability to specify "Chinese New Year" for the first sentence of the post.

Both of the young couple's parents hope the young couple can celebrate the New Year with them, but they don't want to celebrate with each other.

This always confuses the young couple and even makes them to fight;.

Due to the rapid economyic development over the past 20 years, the quality of life in cities has surpassed that in the places where those young people were born.

So they feel very uncomfortable staying where they were born;.

Thirdly, because of the traditional concept of men being valuinged over women, elderly parents are partial to their son's family.

For example , they make their daughter's family do more of the housework, and give their daughter-in-law more smile faces and kind words.

While staying atwith my wife's family , my wife's sister-in-law is always treated better than her, and they say their daughter is always smarter than my daughter. lol.

I wish that elders' conceptideas on these things would change one day!