I’m having a similar problem with Mupen64Plus/Parallel 64, but not exactly the same.
I have two gamepads, an Xbox One controller and a Dualshock 4, but with both of them the problem is identical: I can use the Start, A, B, X, Y, L1, and R1 buttons… but the sticks and digital pad does nothing. For example, if I run Super Mario 64, y can jump, hit, etc, but Mario can’t move.
With any other emulator I don’t have this problem.