I used exactly this multipart parser on my Delphi 2007 project but I needed to adjust it when it came to the unicodification. I guess there could be your problem.
Anyway at least my Delphi 2010 comes with the unit ReqMulti.pas so try to include that somewhere in the project and you should have a multipart parser registered.
If that doesn't work for you I'll post my adjusted parser.