Is there a reason you can't share one $dbh for all children? (I think this is possible with Parallel::ForkManager.)

No, it is not possible.

Reason: Think of two childs, both want to make a query:

# Child 1:
SELECT * FROM foobar WHERE baz = 3

# Child 2:
SELECT grzbaka FROM baz WHERE foo = 5

Now, both send their queries at the same time. The db server might receive:

SELECT * FROSELECT grzbaka FROM baM foobar WHERE baz = 3z WHERE foo = 5

Of course, reality is more complex, this example is just meant as a simple explanation why one db connection used by many childs don't work (as expected).

BTW, if you do cooperative multitasking, like POE does, everything is ok -- there is only one process which sends data to the db server.