Hi,
You are trying to low side switch an active component.
You should be HIGH SIDE switching.
Your config, disconnects gnd from the player, so any other player is still at Vcc, other terminals of the player will now draw gnd current through the Tx and Rx pins, so it never completely turns off.
PLEASE draw a circuit diagram, not a fritzy picture.
What is "P", what are its lead outs.
You need to do something like this, switch the high side, so when turned off the player is at gnd.
To turn the player ON, pin12 LOW.
To turn the player OFF, pin12 HIGH.
Tom... 
PS,This will only work if the Arduino supply and the player supply are the same value.