I grow up in Peru with my parents.
>>>grew for past tense, you could say 'am growing' but that is awkward
My brother was writing about his last vacations.
>>>vacation. If you really want to say plural vacations you should use a word in front like few.
He's looking forward to study Literature.
>>>Studying, and literature should probably be lower case L unless it is a referring to a specific class or type
Vanessa look after so much her dogs.
>>>looks after her dogs so much. Much is awkward here, often would be better
She looks like her parents.
They find out a problem in the laboratory.
>>>is the problem in the laboratory? If so then they found out about a problem in the laboratory. Or is the laboratory where they found out about the problem? - they found out about the problem while in the laboratory.
definitely found instead of find
Clara was in the cinema, when the lights turned off.
>>>turned off is ok, 'went out' is better for lights
The teacher turned on the lights when he came in the class.
She's looking at a traffic accident.
In the market, the merchants put up the lemon's price again.
>>>ok but awkward. Better to say the merchants put up the price of lemons again.
The bus gets in at 8:00 am
I got out the plane when I arrived to Cusco.
>>>I got out of the plane when I arrived in Cusco
I wake up at 6 am every day.
The students stand up when the director of school come in the classroom.
>>>the director of the school comes into. You might want to capitalize Director of the School if it is a title
I went over the list of products.
English isnt easy, and most people would understand what you are saying above, but getting the fine points right helps