That set of symptoms is a new twist (usually turning the parking lights on
will kill the turn signal) but the suspicion still turns to the grounding
for the right front lights. I'm thinking the common ground for all those is
not making contact to the chassis, and the turn signal current is going
backward through the headlights when those are not turned on. If so, the
right headlight will be very dim and the right turn signal (front and rear)
should be on solid when the headlights are on. Otherwise, I dunno, but
grounding is still a prime suspect.

Mike