Why does pp!initiator in Subprocess does not pick the user who triggered the Parent process each time.

pp!initiator behavior in a subprocess depends on how it's invoked. Sync (Sub-Process node) inherits the parent initiator correctly, while async (Start Process smart service) resolves it to the process model publisher instead. Without a detailed process design view, the exact cause can't be confirmed. Make sure you are not running sub process or node as run as whoever designed this process model under assignment tab.

Either way, avoid relying on pp!initiator for user validation in subprocesses;  pass the parent initiator explicitly as an input process variable instead.

In the parent process, you need to check what configuraiton is being used in the "Assignment" tab.  Initiator will inherit down when "whoever started the process" is selected, and likewise if "whoever designed" is selected, the initiator from there down in the process instance hierarchy will be the model's publishing user instead (which is recommended for any process instances that might be open for a while, since deactivation of the initiator user will mess things up).

Apart from that, I second Shubham's suggestion - due to the finnicky and uncontrollable nature of "pp!initiator", you should explicitly store and pass the initiator user as a PV instead of relying on the process property.  The secondary benefit here is, occasionally there will be times (in actual usage) where this needs to be manually edited to a different user, and using this suggestion makes it far easier.