If I relinquish my claim on my dad's life insurance policy, will it automatically go to my mom?

No, and don't do relinquish the claim; if you want to give your money after receipt of payment to your mom that is ok. However when you relinquish your claim there is no named beneficiary, which may throw the matter into a probate proceeding, which may be detrimental to you and your mom; since I do not know the heir structure, some other heir  may make a claim on the money in the probate proceeding. Don't relinquish your beneficiary status.