I have a feeling that all of the child processes should be able to share the same $parent_dbh.
The problem is that when you fork, you duplicate the entire contents of the current process. However, sometimes a process has certain characteristics that are not pure data, but that are tied specifically to the process in which they were initiated. In the case of database handles, the issue is the socket connections established prior to the fork that uniquely relate the process that created them to the database itself. Those sockets can't be treated merely as pure data and hence cloned, because they constitute a relationship between two processes. And that relationship gets broken during the cloning procedure.
AFAIK the only way to fork and share a database handle is to use interprocess communication, or to have some kind of "smart" forking procedure that reexecutes particular pieces of the code that occur before the fork and which cannot be directly cloned. But that would be hard :^P