Dog Sitting now and whyis the dog unresponsive?

Leave him be. He'll go to the door when he's ready. His parents said this is 'normal' for him.
He loves his parents and they have gone somewhere without him and he misses them. So, he's mourning in the only way he knows how.

My dog does the same thing. I left him with the neighbors one afternoon. They have a dog he plays with, as well as a little girl. He is usually super excited to see them. Every hour, on the hour, I got a photo of him in the same spot in front of the door. They tried giving him treats and ...nothing. He wouldn't even turn around to look at them. He wouldn't budge even for chicken.
I don't know how long he would've't stayed but when I got back, 5 hours later, he had not budged.

The dog your sitting will most likely not want to potty in the house, so when it's time, he will (reluctantly) let you know.

All the best

Many dogs don't eat for awhile when they're left with me. It can be a whole day or even longer. I don't get terribly concerned. When the dog is hungry enough, he or she will eat.

I too would recommend that you just let him be. He will eventually come around when he is ready to. You can try asking the owners what other sitters have done in the past? If there is something different that they would like you to do? Does he have a favorite dish you can make him (chicken, beef, can food?) Does he like other dogs? Maybe if okay with the owners bring one of your dogs over to the house (or a friendly dog may make him want to perk up and play) a neighborhood dog that he likes. I would definitely mention it to the owners,they may have a suggestion and if nothing else at least they know that you are trying. Good Luck!