01 corolla, Karr alarm

If the car has 12v+ for your parking lights, leave the red and RED / white wire togather hook into 12v+: The parking light input wire is just the input source to use to turn the parking lights on(If the car had a neg 12v for parking lights, you would ground the red wire.

The purple and brown wires are an either/or type of thing.  If you have a neg door trigger use the brown and cut the purple wire(prob. what you have).  If you have a positive door trigger use the purple and cut the brown.

Don't have to use the domelght supervision wires. 

What kind of alarm is it???

Okay, here we go.  I will list your alarm wire colors first  and your car wire second.

Connect the white parking light wire to the green wire under your fuse box.

Connect the BLACK/ white wire for horn output to GREEN/ red wire in your steering column

The RED / white (12V) wire connects to the white wire in your main ignition harness

Ground the black wire to metal.

connent the brown (-) door trigger wire to RED / white wire in the instrument cluster

the yellow ignition wire connects to BLACK/ white wire in the main ignition harness

All you have left is to cut your starter wire and connect the starter interupt.  From your diagram, you will need a relay to accomplish this.   You can find that info in the relay section of this site.        Good luck and test all wires with a DMM before connecting them.