If you look carefully at the filaments in the left side bulbs that are blinking with the right side signals when they shouldn't, I predict you will see that it is the actually the running [parking] light filament, which is less bright than the signal filament.

The feed to the right front signal is powering the parking lights all the way around. The percentage bet is that your right front socket or wires to the socket have the running and signal circuits shorted together. A missing ground at the RF housing is a remote possibility, but likely wouldn't be a strong enough feedback to power the park lights all the way around.