Some background on the relay circuit that I posted many years ago--it was used with an older version of the prestige alarm/remote start (APS787C if I recall). These older units had a pulse after shutdown that was coincident with the ignition turning off when the remote start shut down. With the 2013 Accent I was working on, the doors would unlock immediately after the ignition turned off (either with the key or remote start), and the parking lights would remain on. The only way to re-lock the doors and to turn the parking lights off was to delay the pulse after shutdown to pulse the lock wire and door pin wire. When this was done, the APS787C alarm would trigger. The timer and relay circuit addressed the problem.
Fast forward to the unit you are using, the APS687TWE. Audiovox updated their E series units with a number of useful features, including a 2 second delay after GWR on pulse after shutdown, as well as the door lock control features in Bank 3, feature 18.
I see you are having an issue with the doors unlocking and the parking lights remaining on after remote start shutdown. You also have a unit without an alarm, and 2012-2017 Accents have no factory alarm. Are the doors unlocking when the ignition turns off? You can confirm this by starting the vehicle with the key and locking the doors, then turning the ignition off and remove the key. Do the doors unlock? The 2013 Accent that I worked on had that issue.
So as Iskidoo stated, no need for the timer/relay circuit. The door lock in feature 18 only locks the doors after start, not after shutdown, so that won't correct the unlock at shutdown issue described above. It will lock the doors after the vehicle remote starts.
You can correct both the unlock at ignition off and parking light issues using the Pulse after shutdown signal from the remote start. Connect 2 general purpose diodes (such as 1N4001) configured as a "Y", with both diode bands tied together and the common banded end connected to the Pulse after shutdown wire going to the Prestige unit. This creates 2 outputs from the Pulse after shutdown signal. One of the unbanded ends will connect to the door locks; the other unbanded end will connect to the door pin. When the remote start shuts down, the door locks will unlock (thanks to Hyundai) and relock and the parking lights will turn off.
Hope this helps.
Lectric Guy