'Three-parent embryo' is a very misleading description. For those struggling with the ethics of it, can I suggest an alternative viewpoint?
Let's say you have a serious medical problem, one that's going to kill you. You can be saved by someone donating an organ so tiny it lives inside cells (a mitochondrion). I assume there would be little resistance to this idea - it's organ donation.
It's quite difficult to get that mitochondrion inside your cells though. You've got bazillions of cells and they all need a donation. Much easier if it had been donated when you were only one cell, then all the cells that are made afterwards will contain this mitochondrion. I assume there would also be little resistance to this idea - it's early organ donation.
So how to get this mitochondrion into your single cell embryonic self? Well, we've got one cell with the tiny organ in it and one cell with the essence of you in it. The essence of you - the nuclear genome - is much bigger (miles and miles bigger) and, importantly, is a single entity. Because we should note that one tiny organ transplant isn't enough - you actually need many mitochondria transferred. So, it's far easier to move the essence of you into the cell containing the multiple mitochondria, than move multiple mitochondria across to the single cell embryonic self.
The cell isn't a mystical entity - it's a bundle of chemicals. And while essence of you will reside, for a short time, inside a cell made by someone else, very quickly you will take over and soon replace all those chemicals with your own (they're exactly the same chemicals but their production will be dictated by you). But those tiny mitochondria, they don't need you to keep them going, they won't be replaced by you. And that's good, because they are your tiny little donated organs keeping you alive.
If you take a chicken egg and a duck egg, remove the chicken yolk and put the duck yolk in its place, what animal will grow? It's not a half-chicken half-duck, it's not a chicken, it's a duck. The cellular environment supporting the yolk (nuclear genome - 'essence of duck') makes no difference to the growing embryo.