ADSL 2 is not very fast and bandwidth is not a lot. Closer you are to the exchange the better and typically the speed of ADSL2 can vary between 500KB/s to 1.2MB/s. More PC's using the same ADSL connection the slower it becomes, your typical copper phone wire has nowhere near the bandwidth of cable or NBN where the latter could suffice 2~3 PC's comfortably all at once with a connection of 50Mbps.
NBN will be much better than ADSL of course and will handle more simultaneous connections in the house without it bogging down like ADSL.
So it has nothing to do with what kind of network card you have its the amount of computers running on it.
For the last 10 years or more, computers have had 1000mbps gigabit nics as a standard feature on motherboards. More than likely your parent's computer has one too. But to reach gigabit (120MB/s give or take) will have to be in a Lan environment copying from one PC to another. NBN is not that fast.
So no, 600mbps nic has nothing to do with it.