It turns out there is more to parsing delimited files with COBOL than a simple UNSTRING statement. There are different styles of delimited files, not just comma separated but also other delimiters and other methods of escaping the delimiter when it appears within the data.
The two resources I used are here and here.
You must know some things about your data before you begin: is there a header record? what is the delimiter? how is the delimiter escaped when it appears in the data? are leading equal signs allowed as an encouragement to accept the data literally?
Once you know these things, you can use the EXAMPLE1.cbl code as a guide to writing your own program to call the CSVPARSE subroutine with the appropriate parameters to parse your delimited file.