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.