First of all, these three institutes are purely science institutes. While both IISc and IISER take through JEE Advanced or KVPY (IISc also takes JEE Main scores), NISER takes through NEST. There are no good institutes for nuclear engineering otherwise as it is a niche field considering that only IIT Bombay offers it and that is a 5 year course. JEE Advanced has to be cleared. IIT Bombay offers Energy Engineering with M.Tech. in Energy Systems Engineering (5 Years, Bachelor and Master of Technology (Dual Degree)) and not nuclear engineering as such.

You parents are right as the class 11 & 12 th marks are important. However entry to good institutes are through entrance tests so you will have to prepare for them. It is not a necessity that you should take coaching.