Must I report my child's foreign bank account if I'm listed as a parent or guardian?

Signature authority is the power to move money from an account by instructing the bank directly, alone or together with someone else. If the bank lets you withdraw from or transfer out of your child's account, you have it. If you are named as a joint owner, you have a financial interest instead. If you can only view the account, you have neither. Filing and signing your child's FBAR as parent or guardian doesn't give you signature authority over the account or put it on your own FBAR.⁠^{[1]Source 1}⁠^{[2]Source 2}⁠^{[3]Source 3}

Your U.S.-citizen child has a foreign account worth $15,000 in the child's name, and the bank accepts your instructions to withdraw from it. You include the account on your own FBAR if your foreign accounts total more than $10,000. Your child also needs an FBAR; if the child can't sign it, you file it, sign it and enter “Parent/Guardian filing for child” in Item 45.⁠^{[3]Source 3}⁠^{[2]Source 2}

Check what the bank's mandate lets you do with your child's account: whether you are named as a joint owner, can withdraw or transfer money, or can only view it. Then work out your own aggregate and your child's aggregate separately.⁠^{[3]Source 3}⁠^{[2]Source 2}