The webseries follows grief-stricken Jason Moore, a teenager whose parents have been killed as a result of a deadly disease spreading like wildfire, caused by a cosmic entity known as the Plague Doctor. Now, he plans to survive in a world where fifteen sixteenths of the population have passed on, and by the end, plans to kill the Plague Doctor, thus ending the Modern Plague once and for all.

A great rewrite by dpg, a logline's purpose is to tell your story in as clear and concise a way as possible. Hope this helps.

A great rewrite by dpg, a logline's purpose is to tell your story in as clear and concise a way as possible. Hope this helps.

What DPG said.

I'll add that a logline for a series, web or otherwise, is best structured to describe the pilot episode. However,?if you do have a the whole season narrative plotted out, you could try and write it for a season but best to focus on the major events in the narrative just as you would in a film - inciting incident?and end goal.

It sounds like your inciting incident is the parents being killed, and the goal is to kill the psychotic entity.
Here is my try at a logline for your season one:
After a psychotic entity kills his parents a fearful teen must fight through a post apocalyptic plague ridden world to kill the entity responsible.