Not sure what country your're in, but there were a few people having issues downloading v1tek's mod. Along with that, other people were only downloading part of his mod rather than all the necessary files (I think on one hosting sites he had to break it into 3 different parts).
Somewhere towards the beginning of the discussion on v1tek's mod is an installation guide, so be sure you're following that closely.
See here:
http://www.racedepartment.com/forum/threads/f1-2013-season-mod.68460/page-7#post-1426936
f.1.
You're not using the right database then, because with my 2013 database HRT has been removed, so there's no way they'd be showing up in a race...unless you have some custom skins or something, or you're using the 2012 database.