Question: Can I balance the cells by simply connecting all the batteries in parallel?
 
Answer: You can do this, but it will not work effectively. Actually you would need a long, long time (days or weeks) to balance the batteries this way.  Effectively said: you can parallel the batteries if you want them all to discharge to the same voltage level over a period of time.
 
However you cannot think this way:  if you have one 100% fully-charged and one 0%-charged battery and you connect them parallel you will get two 50% charged batteries over some time. It does not work this way! 

First: never connect charged and discharged batteries together. Neither in serial nor in parallel.  Use only batteries charged to the same voltage level.

Second: you cannot charge a battery using another battery as a charger. You need a charger to charge. You cannot use a battery to charge another battery. The reason: you need to have a higher potential (voltage level) to charge the battery.