Hello Alina!
The word "permit" means to officially allow (someone) to do something. So "allow" and "permit" are interchangeable in this context.
To "grant" means to agree to give or allow something that has been requested by that person.
So these words are very similar but I think that "permit" is a more formal word than "allow" and "grant" is to give something or allow something after it has been requested.
I hope that helps!