Ok, folks. Figured it out.
There was a report, basically just a few text boxes that were an Action link to other reports. Using that report to get to the other reports somehow screwed up the parameter bar when you would arrive at your destination report.
I removed the 'navigation page' from the reports, renamed the destination reports with friendlier display names for the customer, and all's well. Now they go straight to the report, and there are no issues.