The two Sabertooth members' main point was that Natsu wasn't a true dragon slayer because he never killed a dragon. Their comment about becoming a true dragon slayer after killing their parents could be interpreted as supporting this line of thought as well.
We've never had any real explanation that stated Dragon Lacrimas came from dead dragons, but it could be more easily argued that Dragon Lacrimas can store or expel Dragon magic. Using that logic, one can say that Dragon Lacrimas, while being batteries, are also converters that lets you output dragon-style magic without actually learning how to do it from dragons.
If it's not Isuzu-chan Mii~
That's not how I saw interpreted that scene at all. All those comments to me seemed like a description of how they became 3rd Gen Dragon Slayers.
The implication being that, in order to obtain the Dragon Lacrima they implanted in their bodies, they had to kill their dragons.
If that was the case, then the ones inside 2nd Gen Dragon Slayers would run out after they used all the energy that was stored in the lacrima. And that certainly doesn't seem to be the case with Laxus. The Dragon Lacrima being the very life essence of a dragon explains why they don't ever seem to deplete.