So my right parking light and right tail won't turn off.

I unplugged the battery for 30 mins, they came back on after less than 1 min when I plugged them back in.

I unplugged the SJB and they stayed on, BUT when I plug it back in they turn off; for 30 seconds or so.

Is my SJB screwed? I bought this car from a used dealership five days ago.

 

Take it back to the dealer, hopefully they'll want to keep you a happy customer.