The headlight switch has nothing to do with turn signals
1.....Parking lights ALWAYS went out when the headlights came on in the older cars. This changed, when, about 70?? Look at the diagram of your headlight switch. There is a terminal for ONLY the park lights and a terminal for tail lights. "All Ma did" when she made the park lights stay on, was to move the park light wire over to the tail light terminal!!!
What this means is, your park lights are working exactly as designed. You can change them.......either move that wire, or make a jumper between the two terminals.
2....It is very likely that your turn signal problem is a SOCKET GROUNDING problem.