I'm a little confused as to why you think it will be decided by either parents or 'the government'. Surely teachers and schools will be the first to decide that they want to use online resources as part of how they plan lessons? In the same way that they currently decide to use a textbook, a recorded TV show, a computer application, or other materials.
If pupils are directly interacting with people as part of the process, eg an online tutor, then doubtless that will need to be covered by child protection guidelines. But if they are using online materials then that's not fundamentally different from using written material in a book, except that it can be more varied, and more interactive.
At the extreme end, there are entire online schools, but I assume you are just talking about the use of online resources for specific lessons?
I may be wrong, but given that online learning now has the ability to differentiate automatically for each child's learning it is a bit like comparing the ink pen and printing press.
In America there are now schools where children do most of their learning in subjects like maths from videos at home and work on laptops at school. Teachers then become mentors to help with problems.
Pupils from the poorest areas can have access to the best teaching programs and as a result are making just as much progress as those from more affluent backgrounds. The downside is that this can be achieved with fewer teachers.
Education has been carried out in the same way of a teacher standing in front of a class, using a board to give a lesson since Victorian times. With such big changes on the horizon, it would be nice if parents had a say in the future of education.