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How parthenogenesis evolved is sort of speculative. The first life-forms to spawn on Earth are thought to have been bacteria and archaea that kept re-spawning on their own. Meiosis, the process by which a cell divides into four offspring that each have half the chromosomes of the original cell, probably evolved later. Eukaryotes were most likely the first organisms to use meiosis. Though these types of organisms are unicellular, unlike bacteria, they have organelles and a nucleus. It was most likely a glitch in meiosis that enabled parthenogenesis.
Just in case you are wondering why humans and other mammals are incapable of this, that is because they need certain genes that can only come from sperm. Not that it would stop us from living in a Brave New World type of dystopia if human cloning became a thing.