To begin, it is amazing how many help desk systems do not even include 'parts and billing' in their systems. So 4 stars for WHD for actually having this module.
When you enable Notifications, you should see a Inventory Alert Threshold (which it sounds like you know about), where you can ask for an email to be sent when a part reaches a specific threshold point. FWIW, this has always worked for us.
Underneath Inventory Alert Threshold, you should see an Inventory Alert Interval, which can be set to continue to send emails as long as that Threshold has not been replenished. I have our set at 1 week. So until the Part is replenished, the Default Inventory Alert Recipient(s) will continue to get emails every week. This is set in Setup > Parts & Billing > Inventory Alerts
There isn't a way of knowing which tickets have Parts entered. That would be nice and a good feature request (although new features don't seem to happen in WHD).
You are correct that Parts & Billing is basic, but at least WHD has it. I just wish Solarwinds was more involved with the product to make it better.