Thank you very much for helping me! Here is the lyrics of the song I have to rephrase.

You know parents are the same
No matter time nor place
They don't understand that us kids
Are going to make some mistakes
So to you other kids all across the land
There's no need to argue
Parents just don't understand
I remember one year
My mom took me school shopping
It was me, my brother, my mom, oh, my pop, and my little sister
All hopped in the car
We headed downtown to the Gallery Mall
My mom started bugging with the clothes she chose
I didn't say nothing at first
I just turned up my nose
She said, "What's wrong? This shirt cost $20"
I said, "Mom, this shirt is plaid with a butterfly collar!"
The next half hour was the same old thing
My mother buying me clothes from 1963
And then she lost her mind and did the ultimate
I asked her for Adidas and she bought me Zips!
I said, "Mom, what are you doing, you're ruining my rep"
She said, "You're only sixteen, you don't have a rep yet"
I said, "Mom, let's put these clothes back, please"
She said "no, you go to school to learn not for a fashion show"

Is it OK to say......
Once the singer was taken school shopping by his mom. His sister and and his little brother were there, too. They headed to the centre of town to a large shopping mall. At first, his mother started bugging (pushing for?) with clothes she chose (how can I rephrase it?)
The singer didn't say anything. He just turned up his nose.
She wanted to buy him a shirt which cost only 20$. He replied he didn't like it because it was plaid with a butterfly collar.
It the last straw when he asked her for Adidas and she bought him Zipps. He wanted her to put back the clothes because his rep would be ruined.
His mother replied he was going to school to learn and not for a fashion show. (?)

The only thing I would add would be that his "pop" was there also....

You might also comment about the fact that the song says that this is a common experience in families with teenagers. Look at the first 7 lines. They are the whole point of the song.

Yes, it is absolutely okay to rephrase the lyrics in that way! Here's a slightly adjusted version:

In the song, the narrator talks about going shopping for school with their mom. The narrator's sister and little brother were also with them. They went to a big shopping mall in the city. However, the narrator's mom started to insist on buying clothes that the narrator didn't like.

Instead of expressing their feelings directly, the narrator just turned up their nose, indicating their disapproval without saying anything. The mom wanted to buy a shirt for only $20, but the narrator didn't like it because it had a plaid pattern and a butterfly collar.

Things reached a breaking point when the narrator asked for Adidas shoes, but their mom bought them Zips instead. The narrator felt that their reputation would be ruined if they wore those shoes. They wanted their mom to put the clothes back.

In response, the mom told the narrator that they were going to school to learn, not to show off their fashion sense.