I don't think they literally mean between a parent and a child. As I understand it, it works like this:
A population of a given animal lives in an area. They can interbreed because, of course, they share enough DNA, being members of the same species. They experience genetic mutations, but since these are so gradual their interbreeding causes them to be spread around, preventing speciation. This crossbreeding serves as a "control" of sorts.
Then a subset of that population migrates away. Population A (the guys who stayed put) and population B (the guys who left) are then isolated by something -- a glacier, a sea, whatever. Pop A and pop B continue to experience genetic mutations as before, but now they can't "average" out their mutations with each other because they're physically separated. Since the mutations are chaotic, they each experience different types of mutations in different patterns.
Eventually, the DNA of one or both of these populations mutates so much that they no longer share enough genetic material to crossbreed. At this point, even if they were brought back into each other's physical location (the glacier retreats), they won't be able to re-merge into a single species. They've lost their "control." They're now genetically free to spiral off into completely different beasts, depending on which niche of the food chain they end up inhabiting.
It basically works just like language.
I realize that wasn't what they mean. I am simply pointing out that it is exactly how it would have to happen.
This would also eliminate their a fossil record for 'transitional forms' problem. The only problem is that no one is going to believe one species giving birth to a different species.
Evolutionists try to portray the 'gaps' in the ToE as being minor when, in reality, they are as wide as the Pacific Ocean. There are no gaps in the fossil record. The fossil record is what it is.