Thank you for your reply, Mike!
This sounds promising, but for my rest service I only have the option of Query, Update Features, Apply Edits, or Generate Renderer.
(Create Replica is not a choice)
Is there something different I should do when I first set up the parcel service? Our organization's administrator actually posted the data, but I could take that part over if it gives different access rights.
I've attached a screenshot of the rest service page that I come to for my parcel layer.