Parallelizer never sends anything on its own. It keeps a small local log (no layer names, file paths or coordinates) and, when you choose Report a problem…, shows you a report you can read and edit before deciding what to do with it.
Software versions (plugin, QGIS, Qt, Python, GEOS, PROJ, GDAL), your operating system, locale and screen scaling, other active plugins, the plugin’s settings, the last error, a description of the layers, project and features the last few operations worked on (types, coordinate systems, counts, validity), recent plugin log lines and, if QGIS crashed or froze last time, the Python stacks it left. Paths to your home and QGIS profile folders are replaced. Geometries are only added if you tick Include the geometries involved, and then they are moved to a local origin so their real location is not included. You can edit or delete anything before sending.
Reports sent with Send report are deleted once the problem is handled, and at the latest after 12 months. The dialog shows a reference number after sending; to have a report deleted earlier, or to ask what was received, open an issue on GitHub quoting the reference (don’t include the report itself), or write to the email address shown in the plugin’s report dialog.