I can guarantee that any normal parent and school official would respond ABSOLUTELY NOT! No teacher or adult should EVER tell a student to keep anything from his/her parent! That sends a huge red flag for me, and I would bring this to the principal's attention immediately!
I would say that no, this is NOT normal. To be very honest, "sneaky" tactics such as these tend to be ignored or not seen as an issue. I would contact the superintendant of your school district, before contacting the principal. Both parents and teachers should always promote a trust in each other, not specifically tell a child to ever keep things from a parent.