It seems highly unlikely that a species would evolve in such a way that they would require three or more parents to reproduce, barring highly specific selective pressure.
A scientific paper analyzed why only two sexes evolved using mathematical models. The models showed that three-sexed systems are less responsive to selection pressure than two-sexed systems, and have more genetic variability after the same amount of time under that pressure.
In order for three-sexed organisms to compete, they would have to evolve in conditions that change too rapidly for their slower evolution to matter and which frequently force them into population bottlenecks. The advantage of three copies of the genome from three parents means that inbreeding depression would be much less of a problem.
The scifi novel
Silent Runners used a tri-sexual model for its alien race.
The hydrans in the
Star Trek EU appear to reproduce this way.
Species that reproduce through sexual parasitism (e.g. hybridogenesis, gynogenesis, androgenesis) may be said to possess multiple sexes, but this is on the demographic level rather than the individual level. Obligate sexual parasitism has been observed in amphibians, reptiles, fish and insects. It may occur in other species as well, but we wouldn't be able to tell without genetic sequencing.
This is a completely different concept from a species which is isogamous. In fungi, some algae and some microorganisms, there are no males and females because they produce identical homogeneous gametes (or no gametes at all). Sexual compatibility is determined by mating types (as the name implies, operate on the logical reverse of blood types), of which there may be anywhere from two to many thousands. Although such organisms lack male or female sexes, they still only require two parents to reproduce.
Although limited to non-animal life on Earth, there's no reason why there can't be isogamous megafauna in a fictional setting. I can't actually name any examples except for
this deviantart gallery.
Even within organisms limited to male and female sexes, there is still a fair potential for variety.
According to ScienceDirect, "There are six major sexual systems that involve hermaphroditism: simultaneous hermaphroditism with obligate selfing; simultaneous hermaphroditism with outcrossing in both roles; sequential hermaphroditism, androdioecy, gynodioecy and trioecy."
Even within a species limited to "males" and "females", there's no reason why there couldn't be multiple biological castes limited to certain genders. E.g., a species might consist of seven castes: five males, one female and one sterile/sexless.
E.g. the Dirdir from the Jack Vance novel of the same name: they have 12 types of "males" and 14 types of "females." No details are given.
Not to mention the diversity of sex-determination systems. E.g., a species could spawn males asexually and females sexually.
Or, I don't know, a species could reproduce in some alien fashion where the females are born pregnant and when the male copulates with the female he actually inseminates the offspring in her womb. (I got the idea by looking at the telescoping generations of aphids, then switched the generations of females but not the males!)
Or an animal species could alternate generations exactly like plants!
I don't think scifi has scratched the surface of our imaginations yet.
