Unless the PS3 slim is soft or hard modded, no you can not play PS2 games on the slim PS3 or later models. The only PS3 system that can actually play PS2 discs is the original 20 40 gig models as that had backwards compatibility.

The PlayStation 4 disc drive and hardware can't read PS2 or PS3 discs, so the easiest way to access your favorite old games is to use a PlayStation Plus Extra or Deluxe membership. The Extra tier provides access to the Game Catalog, which includes some PS4 titles.