Along with your son's teacher's interpretation, there is another interpretation if you just look at the grammar.
When describing a tennis match, for example, you could say:
I played John McEnroe in the finals.
In this usage, against is missing but understood to exist.
So, if we take your sentence, how it could be interpreted is:
We played [against] six balls at the same time.
Again, if this were a tennis match, it would mean that you were on one side of the net and you had six (animated and tennis-racket wielding?) balls on the other side of the net.
Note that there's nothing wrong with the grammar of your sentence, simply that it doesn't imply with in the same sense that you want it to. Either it means you played the game six balls (as the teacher indicated) or that you played against six balls (which makes little sense).
Also note that even if you do explicitly use with, the meaning could be ambiguous. It can either mean playing alongside someone (John and Mary played bridge with Martha and James) or it could mean an item being the object of play (the children played with the toys).
In the case of we played with six balls, the meaning would be assumed from context— and the fact that balls aren't sentient and can't be partners, meaning they should be the object of play.