Just because one parent pays child support and carrys insurance on the child do they have the right to claim them at income tax

No, the custodial parent must relinquish the right to take the deduction.

This response does not constitute legal, accounting or other professional advice. Only through a personal, confidential consultation with qualified legal counsel can anyone properly evaluate their own unique legal challenges and determine what, if any, appropriate legal strategies and tactics should be implemented to meet those challenges.

Circular 230 Disclaimer - Nothing in this response is intended or written to be used, and cannot be used by any person for the purpose of avoiding tax penalties regarding any transactions or matters addressed herein. You should always seek advice from independent tax advisers regarding the same.