Parking Light And Signal light with same LED using MOSFETS
As the schematic is drawn:


When parking light is on and signal is off


Q2 will conduct through its body diode.  Since there is no safe current limiting component, the weakest one of Q2, D1, and LED will blow.


When parking light if off and signal is on


Q3 will turn on. Since there is no safe current limiting component in that path, the weakest one of Q3, D2, and LED will blow.


When parking light is on and signal is on


A combination of the above will occur.

This is a bit tricky due to the third bullet item requirement.  My suggestion would be to create a truth table.  Could this be an XOR?  If so, it could be handled by one gate, one FET, and one resistor.