You need to send pre-authorizations to both companies. When claiming for a child, each month, the parent with the birthday that falls first in the year, would claim first, and then send the remainder to the second or third company along with a copy of the explanation of benefits. You cannot claim more than 100% of what you pay.
Simple rules—always submit the claim first to the benefit plan that belongs to whoever received treatment and to the other benefit provider once you have received your claim payment and statement from the first.
With coordination of benefits, you claim first with the plan you are a member of, then you claim any amount that was not reimbursed with the second plan.