When Epsilon was not yet mobile, and my partner was traveling, I would call my mother on skype, and put the computer down next to Epsilon while he played on his blanket with his toys. This let my mother talk to her grandson, kept Epsilon vaguely interested in the funny faces and sounds she was making in his direction, and left me free to clean the kitchen, knowing that my mother would call me if Epsilon needed my attention. It was a kind of remote babysitting. Kind of. But it was a win for all involved.
Now that Epsilon is a toddler, my partner Skypes me from two time zones away when I'm done teaching, and it's time for their dinner. And we have a "family" dinner. I recite Epsilon's favorite stories to him, with or without the relevant book in his hand. He tells me in no uncertain terms that Mama's not there. Or his father maneuvers the computer around the apartment as he shows me his toys.
I had a colleague visiting my university last year for a month, when he had a child about Epsilon's age. I would catch him finishing up a Skype session with his wife and child as I stopped by his office to meet for lunch. He would tell me stories of his son trying to share with him the biscuits he was having for breakfast. Sweet yes, but...
Parenting by Skype isn't really parenting. It doesn't actually provide any relief for the parent on the other end doing all the work. I'm not even sure if the benefits of seeing the rest of my family outweighs the heartbreak at the end of each phone call.
The aforementioned colleague will be visiting again for a few weeks soon. I suspect there will be some quality time spent over good alcohol.