It's either one of two things ... an inadvertent MIDI mapping assignment to the play/pause of the right deck (mapped to the left play/pause button) or you've got faulty hardware. They're both easy to check. For the first, open SDJ with the SR connected and click the MIDI button. Click the PLAY icon for the right deck and verify that no controls are mapped there. You can press the DELETE key to be sure.
To check if the button itself is working, go grab MIDI Monitor (free) and check to see if the left play/pause button is sending signal - if so, what signal is received. Next, press the right play/pause button and check what signal is received.
Let me know what your findings are.