Of course everyone is concerned about child safety, but it doesn't follow that you have the right to do anything that increases child safety without taking into account the impact it has on others.
You could argue that you neighbours shouldn't be able to have a dog because it may kill or hurt your child or that any visitor to your strata has to be police checked for prior convictions of violence to children. You have to balance the benefits to you with the impact it has on others. Child restraints in cars should be compulsory because it saves lives and doesn't impact negatively on others, plus people have to put their kids in cars to transport them.
The Strata is concerned with the visual impact of the nets and the negative affect it will have on the other properties and the physical effect on common property(fixing the nets to the structure). Parents aren't being forced to live in a property that is potentially dangerous to their child.
So I think that if the consensus of the other property owners is that they don't like them, then so be it. They shouldn't be held to emotional ransom by the excuse that it's to protect someones kids.