The asymmetry of time plays an important role in the personal identity debate. Since the past cannot be recovered, we have no interest in it. However, the future is required for continued survival. So, the relation of the current individual to his/her/its future self, is vital, and the relation to the past is incidental. Nevertheless, let's see if the relation to the past sheds any light on personal identity.
In the example, the old woman's body and mind has been produced from the ten-year old girl. The old woman has memories of the ten-year old from an inner point of view. She remembers being the ten-year old girl. These memories aren't exact, they are partially reconstructed from whatever memory traces are left with what she imagines it was like in order to weave a coherent but not necessarily rational recollection. Parfit points out that if we could transplant memories, we might share experiences with friends by letting them experience it through a transplant. Such a transplant would not transform the host into someone other than what he/she/it is. The host would know who is who, and remark, "Oh my what a lovely time you had" to his generous friend as he recalls the memory.
Now, let's notice what happens when we set aside the woman's memories. First, the class of examples widens to encroach on the question of the identity of changing objects over time, in general. For example, a watch has a new faceplate and mainspring. Its parts have changed, but if it was Sam's watch before the repair, it still is his watch afterward. Sam's memories and the repairman's memories are required. However, these memories are extrinsic identity conditions and so succumb to the charge of being merely a social convention about who is going to use the now changed watch. What intrinsic criteria coheres the identity of the watch? Or, is it always just a collection of parts designed to work together for a particular function, and there never really is anything there beyond that. All we really conclude on this last scenario is that the repair does not compromise function.
Adequate repair does not compromise function. This principle will become very important when we consider further what matters in survival. Matters relating to memory, or to epistemological criteria of identifiability are secondary. The little girl survives to be an old woman through her causal continuity, not her memory.