You have to use the balance plug to balance the batteries, it can't be done through the EC3 connector.
No your parkzone battery does not have to be charged/balanced with parkzone charger. Any quality charger/balancer will work fine.
There are 2 different size balance connectors (not talking about pin count but physical size) one is Thunder Power, the other I am not sure of the name but most other battery manufacturers use it. The second, larger size, is probably what fits the Equinox balancer and probably what you have on your batteries now. You would probably need an adaptor for Thunder Power batteries (the polarity is also reversed on Thunder Power)
Notice all the "probably"s in my post.