New York does not allow joint account ownership, so if your parents file separately, they will each need their own account. If they file a joint return, they can have a single account and pool their contributions so that they both get the tax break. You can find details here under, "Tax Benefits."
It is usually easiest to make the parent the beneficiary to avoid any generation-skipping tax implications, which can occur when changing beneficiaries more than one generation apart. It may also make sense to transfer ownership of the 529 plan to you once the future child or children have a tax ID and can be assigned as beneficiaries to the account (see this article). It all depends on your and your parent's financial situation, though.
VP, Research & Development
This information does not constitute tax advice and is provided for informational purposes only. Please consult your tax advisor, financial advisor, local taxing authority, and/or plan provider or sponsor for more information.