I don't know if I'll ever deal with the details of the death of Tim's parents in the webcomic. I'll say enough that they both died at the same time, leaving this Tim orphaned at a young age, even younger than Bruce's was (Tim's about six years old).
I see the Cabbage Patch story as a way for Tim to cope with his loss and to help him make a connection with his new family.
Some background on the character Tim Drake:
In the DC continuation, both his parents were murdered. Janet Drake was poisoned by Obeah Man and a few years later, Jack Drake was killed by Captain Boomerang.
In the animated series, Tim was amalgamation of Tim Drake and Jason Todd of the DC continuation, giving that Tim's origins similar to Jason's. Tim's father was known as Steven 'Shifty' Drake and was a small-time criminal who worked for (and was killed later by) Two-Face.