It is decently possible that it is an electrical short making it register the mist setting instead. A test of this would be to turn it to the mist setting to see if it turns off, as shorts can cause triggers to switch.
You simply cannot. Sorry.
Bad door switch or bad headlight switch
Many vehicles have a "mist" feature that makes the wipers swipe one time upon startup to remove mist or fog. This normally can be turned off...check manual.
wiper motor could be going bad
It sounds like you are not turning the wipers off by using the wiper-switch, but are instead just turning off the ignition-switch. The result will be as you described. Try FIRST turning off the wipers with the wiper-switch, and THEN turning off the ignition-switch. Wrong answer , the question is when the --wipers are turned off- -they stop sticking straight up . not when the ignition is turned off . Delay wipers have a relay module under the dash and above the brake pedal. It's a black plastic box approx 4"x6"x1" with a connector on the short end. The park relay in that module is defective, but as it's on one circuit board, you have to replace the whole thing. it ran me $75 from Ford/Lincoln.
change your wiper switch located on the steering column
Gremlins
The wiper motor has a switch inside of it. When they are turned off and the wipers go to the down position it trips the switch in the motor that tells the computer they are in the down position. If that switch in the motor is faulty, the wipers are repositioning until the switch is tripped to the off position. THE FIX.... Replace the wiper motor.
replace the wiper motor
What does a remote starter have to do with the wipers? Just turn them off when stopping the car, and then when you use the remote starter, the next time, the wipers will not come on. If the wipers are not on then they will not work, no matter if you are using a remote starter or not.
lights must be on first off....then when your normal wiper is on push in the washer fluid control....the headlight wipers should come on then...there is no seperate controller for them
Wiper motors have an internal switch that keeps the wipers running after you turn them off. The switch stays active until the wipers get to the "PARK" position and if the drivers wiper switch is off, then the wiper motor is turned off internally. I suspect the internal switch has failed. Unfortunately the switch cannot be easily replaced. Fortunately, wiper motors are not terribly expensive.