The article doesn't really explain, very well, the distinction between mitochondrial DNA, and a human's main DNA; and an understanding of this distinction is crucial in order to understand what the article is really about. What is being discussed, here, is not really nearly as bizarre as it sounds.
Some time very early in the evolution of life, a bacterium that got eaten by an early Eukaryote, instead of being digested and destroyed, instead started living symbiotically with that Eukaryote, providing additional benefits to the Eukaryote. The mitochondria in our cells today are descendants of that bacterium, having their own DNA separate from ours. Of course, in all the billions of years that Eukaryotic life has existed, these mitochondria and other organelles derived from ingested bacteria have evolved to fulfill their roles, not as individual bacteria, but as functioning parts of a greater Eukaryotic organism.
What the article is about is people whose mitochondria are in some way defective, and of a proposed means of preventing it which involves splicing mitochondrial DNA from a different donor into a human zygote, on place of that which one would normally inherit from one's mother.
I disagree with the terminology that suggests that the mitochondrial donor would be a third “parent”; and with the bizarre ethical implications that are suggested by this terminology.