Yes, you can add multiple parents to a child fairly easily. If you set one of them as the billing contact then that parent will receive any emails related to invoices/payments, etc. However if both are set as the billing contact, then both will receive emails for payments (as well as other messages sent through the system).
The only way to ensure one parent receives certain invoices and the other receives others - is by turning on and off the billing contact setting for each parent.
Unfortunately the system can't tell which parent would need the billing information without you doing this.