Is there any way to possibly appoint my parents as proxies in my place?
I don't see why not.

Provide them with all of the check images you have showing the amount paid. They'll also need that copy of the lease you've got lurking about, and it would be helpful if you still had the welcome packet as well.

Did you receive a receipt or itemization when you received your security deposit back? Fish that out, too, if you can.

Your parents can then send copies to your former landlord.

It's entirely possible that the landlord simply made a mistake in his record keeping. It happens.