Re: Are there any species on Earth that have three parents?

No, there are no naturally occuring species on Earth that can have three or more parents, but then there are some almost exceptions. There are actually several species of bacteria, protists, algae, and fungi that have multiple "mating types", or more than two genders; however, only two individuals of different genders at a time can produce offspring. Since I am more familiar with fungal life cycles, I'll use them as an example.

Fungi use two methods of reproduction, sexual and asexual. Many, if not most, fungi spend most of their lives as haploid (having only one set of chromosomes: n) organisms, referred to as anamorphs , and reproduce asexually by forming spores (also haploid), which disseminate to form more haploid fungi. At times, these haploid fungi come into contact with other haploid fungi of the same species, and if they determine that they are compatible, they fuse to enter the sexual reproductive cycle as teleomorphs. In chytrids and zygomycetes, this means forming a diploid (having two sets of chromosomes: 2n) fruiting body that can then undergo meiosis to form new haploid spores. In the basidiomycetes and ascomycetes (collectively referred to as "dikaryomycetes" or "higher fungi"), the fusion of two haploid individuals' "cells" (cytogamy) does not include fusion of their nuclei (karyogamy), and instead results in each "cell" containing two haploid nuclei, a state referred to as either dikaryon: "two nuclei", or heterokaryon: "different nuclei". Most basidiomycetes prefer to live as dikaryon teleomorphs, while most ascomycetes prefer to live as monokaryon anamorphs; however each can live as either form. When conditions are right, the dikaryon fungus can produce dikaryon fruiting bodies (the large, visible dikaryon fruiting bodies are called mushrooms) that contain specialized sexual cells, which undergo karyogamy followed immediately by meiosis to produce spores.

I've gone through all of this, because the measure of sexual compatability between fungi of a given species is determined by the products of one or two genes, and for each of these genes there can be tens to tens of thousands of alleles, which translates to tens to tens of thousands of "genders" (mating types). As with genders in plants and animals, individuals of a given mating type cannot reproduce with individuals of the same mating type, but there are also preferences of each mating type to complementary types. The mating locus genes code for pheromones, and it is through these pheromones that fungi choose their mates; in fact, by serendipity the most scientifically studied ascomycete, Saccharomyces cereviseae or Brewer's yeast, happens to have only two mating types, a and a, which have allowed researchers to identify the mechanisms through which these genes and their pheromones interact. This is relevant to your question, because a single anamorph can form dikaryons with multiple "partners" simultaneously, such that the whole fungus can be composed of more than two different haploid nuclei. As well, there is a certain amount of "parasexual" recombination in the dikaryon (pictured here), whereby the different nuclei can fuse, recombine, and divide through mitosis instead of through meiosis in the fruiting bodies.

Outside of naturally occuring lifeforms, there are engineered organisms that can have more than two parents, called chimeras. A chimera is an organism that is formed by mixing the cells of two (or more) early-stage embryos, and then allowing the new mixed embryo to develop. This has been done extensively in mice (pictured here). The final chimeric mouse is formed from the cells of both of the original embryos, and each of the original embryos was formed through sexual reproduction between two parents, so as long as all of the parents are different, a single chimeric mouse has, essentially, four parents. (In this picture, the coat colors of each set of parents was specifically chosen to demonstrate the different parental origins of the different patches of skin.) However, at the cellular level, there are still two populations of cells derived from each original embryo, with no fusion or recombination between these cells. As a result, each offspring of a chimera is really the offspring of one of the two original embryos, but never of both.

The offspring of the chimera and the fruiting bodies of the fungi point to the answer (finally) to why sexual reproduction requires two, and only two, parents: meiosis. Meiosis is the cellular mechanism, through which haploid gametes are formed from diploid sexual cells. During meiosis I, the chromosomes of the diploid progenitor cell line pair up with their homologous "sisters", and then each goes to opposite cells. The problem with having more than two parents, especially an odd number of parents, is to which cell do the extra chromosomes go? With three parents (and thus a triploid genome), the third set would be distributed randomly, generating unbalanced chromosomes, which is usually lethal to the developing gamete. If nothing else, the offspring produced from these unbalanced gametes would have different chromosome numbers (karyotypes) than their parents. Although several species of plants are tetraploid (4n) and octaploid (8n), these result from incomplete division during meiosis I rather than from multiple parents. Also, tetraploid plants can only fertilize other tetraploids to produce fertile offspring - each tetraploid genome is treated as a diploid genome during meiosis. From a simple numerical perspective, meiosis is unable to cope with multiple parents, since each round of offspring would end up with bigger genomes than their parents, e.g. if each of four diploid parents produced a haploid gamete, their combined fusion would produce tetraploid offspring, each of which would produce diploid gametes, four of which would fuse to produce an octaploid offspring with tetraploid gametes, etc. The only way to overcome this would be for the organism to devise a completely novel form of meiosis, using a different cellular machinery, to divide each sexual progenitor cell as many ways as there were parents.