Who claims the child as a dependent on their 2024 taxes the parent that has custodial rights or the one that pays support

Are you the custodial parent?  Do you have an agreement with the other parent to allow the other parent to claim them--due to divorce or that you live apart and share custody?  Did one of you sign a Form 8332?

If there is a signed 8332 then the custodial parent retains the right to file as Head of Household, get earned income credit and the childcare credit.  The non-custodial parent gets the child tax credit for children under the age of 17.  If the child is 17 or older the non-custodial parent gets the $500 credit for other dependents.

If you and the other parent have a signed agreement, you need to indicate in MY INFO that you have such an agreement.

As far as the IRS is concerned, the custodial parent is the one with whom the child spent the most nights during the tax year--at least 183 nights.  If there is no signed agreement, then the custodial parent can claim the child for all child-related credits including child tax credit.