The child should file the 1099-C because the child is the debtor. The co-signing parent is a guarantor not a debtor according to the IRS (https://www.law.cornell.edu/cfr/text/26/1.6050P-1)
The co-signing parent should not have gotten a 1099-C and you should contact the creditor and ask for a corrected 1099-C. If the creditor refuses to correct the 1099-C issued to the co-signing parent, the cosigner will need to attach a note to the tax return with an explanation.
