24V LED strips connected in parallel - How should the data line be connected?
You are correct -- you should leave the ground and data lines (black and green) attached on the JST connector, but disconnect the power line (the red wire).

What would actually probably happen if you left the red wires connected, is that the different power supplies would "fight", and it would probably be harmful to them, depending on their design. Internally to the strip, the two red wires are connected together, and the two black wires are connected together. So if you don't disconnect the red wire from the JST connector, you will just be connecting the different power supplies' outputs directly together.