You can’t configure DNS server by name on anything, so you’d need some kind of script/automation to query current IP address of your pihole from google/your ddns provider/someone and update that on your parents router which can be a bit tricky or straight impossible depending on the hardware.

VPN would solve both 1 and 2 from your list as your pihole would be available with static address on both locations. You can’t authenticate on DNS server by MAC as you don’t receive originating MAC at all. Other solution would be to get a static IP address from some provider and tunnel traffic so that your pihole could be reached trough that static address.