Parthenogenesis is possible in humans but very unlikely to result in a viable baby. In order for an embryo to develop from an unfertilized egg, the egg would have to sense a spike in calcium, skip meiosis and then lose at least two specific maternal genes.
When an egg is fertilized, it loses half its genetic material, replacing it with the sperm's genes. Then it receives specific genetic information from the sperm DNA to ensure proteins are produced in the correct quantities, a process called imprinting. If imprinting does not work properly, the embryo's cells will start dividing but it will die within days. The chances of all these things happening properly are minuscule; by one estimate, the chances of altering just one specific gene are about a billion-to-one, and in a human, about two hundred genes must be changed.