Are only the park lights supposed to come on when the switch is in the first position?
Are the park lights supposed to turn off when the switch is in the second position (headlights on)?
All the lights work (i.e.brake, turn, hi-beam,low-beam). If I leave it the way it is, will it cause any problems?
Probably OK. If the circuit breaker trips, you have too much load. I wired the parking lights to the tail light terminal so the parking lights are lit in both positions. No issue with that.