There are 3 problems that you're reporting.
1. When the 4-way flashers are active and the brakes are on then all 4 bulbs remain lit instead of flashing as one would expect. I think this may be normal and is a function of the blinker/hazard switch in the steering column.
2. The dash blinker LED indicator bulbs will come on when the parking lamps are on. This obviously shouldn't happen and I think the problem is caused because the parking lamps are back feeding voltage onto the blinker circuit when the parking lamps are turned on.
3. When the engine is running and the parking lamps are on then the front blinkers will not function. So again I think this is the same as problem number 2 but it is more pronounced when the engine is running because the operating voltage is greater than when the engine is off. It could be that the blinkers actually are blinking but you can't tell because the back fed voltage from the parking lamp circuit is enough to not only light the dash indicator bulbs but it is also now lighting the blinker bulbs.
So this seems to be a defective digitails parking lamp or lamps. You could try probing either the left or right blinker wire going to the digitail light and see if a voltage is present when you only turn on the parking lights.
I suspect that you will find a voltage on both the left and right sides since you stated that both indicator bulbs are coming on. You could work around this problem by adding a diode on each blinker wire going to the digitail parking lights.
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