I don't know exactly what caused the problem (most likely a glitch last time I updated the Plesk software), but the problem was very simply a couple of blank fields in the database. It was turned out to be incredibly easy to fix.

Step 1: accessing the Plesk database

1. Login to Plesk as admin
2. Access the "Database Servers" section (there's a link to this on the "Home" page of my control panel, but this may not be the case on all setups)
3. Click "Local MySQL server"
4. Click the "Databases" tab
5. Click the "Webadmin" button at the top
6. Click the database called "psa"
7. Click the table called "dns_recs" (on the left-hand column)
8. Click the "Browse" tab

This solution worked for me as admin, but it may not work for every installation. Unfortunately, the root MySQL password is not a piece of information I have for my installation (it's buried in Plesk somewhere), but if you know yours then that's a viable alternative if you can get phpMyAdmin or similar setup.

Step 2: modifying the data

In the "dns_recs" table, you'll hopefully see a number of DNS entries. You should see one or more with a type called "PTR", and you're looking for any such record where there are blank spaces for the "displayHost" and "host" entries. In my case, it was just the very first record that was a problem. I was able to copy the "displayHost" and "host" values from a different PTR row, and insert them into the problem row.

In my case, the missing value was just the IP address of my VPS (I only have one IP address). In your case, it may be different, especially if you have multiple IP addresses. I recommend trying your main IP address (if you have one).

Done!

Once again, I'm afraid I don't know exactly what caused the problem, but that seems to be the solution -- fix the empty data in the dns_recs table. If you find a different (perhaps better) solution to the same problem, then please share it in the comments below!