No, that's not true. If you have told the program that you let the other parent claim him, it does not mean you claimed your son and the other parent cannot file to claim him. With that permission, the other parent can claim your son on his return because you release your right to the father to claim him on his return.
Just make sure you do these steps in the program under the "Dependent" section. Please see the attached screenshots.
As you are the custodial parent, with the release, you can claim Head of Household, Dependent Care Credit and EIC and the other parent who is the non-custodial parent will claim the dependent and child tax credit.
