Why पर not में?
में could also work.
The latter sounds better to me too.
My Hindi teacher told me both "men" and "par" mean location or place. Often "par" is used in same way where "at" is used in English, and "men" where "in" is used. Here it's exactly the case - "at home" is "par ghar".
Said no Indian parent ever.
घर पर मत पढ़ो sounds strange - it literally means do not study ON the home. घर में मत पढ़ो should be the better answer.
Does पे work instead of पर?
Yes. They are interchangeable.
I dont know why they use such sentences like billi pur baito and gar par math pado are used. Sentences of such negative thoughts should be avoided.