In making such decisions as whether to grant parole, should we care whether
I think so. After all, even if remorse is not an absolute guarantee that the remorseful person won't repeat his/her wrongdoing, it is at least a positive indicator.
There are several theories of punishment and so the very idea of parole will vary under different theories. For example, in a retributive theory, the main question will be whether the criminal has "paid his/her debt to society," and it would seem that, strictly speaking, this could simply be a matter of doing the time in prison or whatever. On the other hand, I don't see why a retributivist couldn't think that part of the appropriate "price" includes feelings of remorse. In a social protection theory, the goal is simply to make sure the criminal is no longer a threat. It would seem that his or her feelings of remorse would be at least one useful indicator of whether he or she continued to pose a threat of the relevant sort. The same goes for a rehabilitative theory, where remorse might reasonably be taken as an indication off rehabilitation. For the theory that punishment is supposed to deter crime, however, it is not clear to me where the notion of parole would come in at all, in which case the question is moot under this theory--though I am probably missing something here.
Anyway, the point is that under most general theories of punishment--or at least those that see a role for the practice of parole--there is some reason to see remorse as a relevant and positive factor in favor of parole, though it would not be the only relevant consideration, I think.