From the pictures, the current wiring has the taps (green and white wires) necessary for balance charging and monitoring. The power cable goes to the board, not the batteries directly, so chances are the circuitry is in full control of charging. The manufacturer's info for that pack (2B0500042) claims it's protected (overcharge/overdischarge/overcurrent/reverse input/short).
alpg88 is right -- do not use protected batteries or otherwise add circuitry to that pack. Without a full understanding of the design and behavior of the existing circuitry, you cannot predict what will happen when adding more.
The PCBs on protected batteries work by looking for a specific set of conditions and then breaking the circuit when they detect something they don't like. (Note that "what they don't like" is external -- protection won't stop a bad battery from internally shorting and bursting into flame, it will just stop the load from pulling more current than it's supposed to, or the charger from trying to charge too long, etc.) When added to a pack like that, the PCB triggers may not match the load characteristics needed for normal use of the pack, and trip when they're not supposed to.
When one of the protected batteries trips, suddenly one of the two parallel cells goes "missing", leaving the remaining cell to take the full load or charge. At minimum, this will increase stress on that cell, and the sudden change in state at that tap may cause the pack's board to behave strangely. The behavior of a tripped protection circuit on an individual cell does not match what happens when the cell is wearing out, which is the situation the pack's board is designed to handle. Also, once protection trips on a battery's PCB, it needs to be reset to use it again -- but you may have a hard time doing that while it's part of a pack.
And that's without considering what might happen when one of the protection PCBs fails in some strange way...
If properly designed into the entire system, redundant protection would be fine, but it's not something that can just be added to an existing design after the fact.
One thing you should do to maximize safety is use high-quality cells as replacements (and make sure they match or exceed the charge and drain ratings of the current ones, of course). One of the most important things in a pack is that all of its cells behave as consistently as possible, and with the very cheap cells there's a greater concern of variation.