Getting Type 1 font support error message about non-existent Type 1 fonts

I only have TrueType and OpenType fonts installed and I'm working with templates and documents created using only TrueType and OpenType fonts, but when I open files in InDesign, I get that big purple banner across the bottom warning me that my document contains anywhere from 1 to 30 or more Type 1 fonts and instructing me to replace them (without telling me which Type 1 fonts it says are in use).  I've gone through every paragraph style and character style in those documents to confirm that only TrueType and OpenType fonts were used and the Find/Replace font tool only shows the TrueType and OpenType fonts.  

I haven't used PostScript fonts in more than 15 years and none of the five computers I've used in the last 15 years (to create these very templates and documents) have ever had the older PostScript Type 1 fonts installed on them, so I'm baffled about what's causing those alerts.  

The only time I don't get those errors is when I open a new blank document.  Using those exact same TrueType and OpenType fonts in a new document does not generate errors about Type 1 fonts being used.  This only happens with legacy documents created using older versions of InDesign.

Any idea what's causing this and how to resolve this error message?