I went to convert a file that should be a small box and it includes several parts that definitely do not belong in the assembly and they are all suppressed with a missing link to the files for the suppressed parts. 
I would like to know why these parts are being put into each parasolid file I try to open, and how to stop that from happening.
I have similar issue - whenever I export any assembly to Parasolid or STEP, then open the exported file back into Solidworks, there are several files listed that are suppressed links to missing files. I don't have these files anywhere, and they weren't in the original assembly. My assembly template is empty. I can delete the files, save as an assembly, open that assembly, export the parasolid again, and the broken file links re-appear when I open the parasolid.
Check which assy template is specified in this location. Check and make sure that template is empty/correct.
that was it - the junk was in the template. Thanks!
Check you assembly template file in case there are already missing components in there. If that is OK then can you share one parasolid file to investigate.