How would such a system work, a) physically, and b), at the cellular level? (we need three different types of cells: sperm, egg, and ???)
Nutrition. To reproduce there would need to be genetic material from a sperm, an egg, then that fertilized cell needs to find a place to get nutrients to develop until it has the means to get nutrition independently by hunting for food, photosynthesis, or whatever.
Consider a species that has a male gene I'll call "M" that is somewhat analogous to the Y chromosome, and a female gene "F" somewhat analogous to the X chromosome. Instead of producing XY male and XX female it there's MM males, FF females, and MF "hosts" that provide the nutrition needed for development.
In this case the species would produce children that are 1/2 MF, 1/4 FF, and 1/4 MM. All the children are born in an infantile state but only the MF children are able to fully develop into a sex that is large enough to have a complete digestive tract and live independently. The MF sex is effectively an asexual being upon which either a FF or MM will attach to provide, or effectively become, sexual organs.
Maybe the MM and FF sexes of the species are little more than a kind of parasite that provide the reproductive organs for the MF host sex, their bodies kind of "dissolving" into the host except for the bare minimum needed for reproduction. I recall that there's some fish species that do something like this.
Maybe males and females develop to a kind of child that becomes fused to the host in a kind of pouch. While it can see and/or hear, and communicate with the host by movement, hormones, simple verbalization, or something, it lacks the mental capacity to know or feel much but pain, fear, pleasure, and so on. While they provide the eggs or sperm to the host they can also provide sensory organs to detect hazards and/or potential mates.
Perhaps the males and females are peers to the host sex, creating a creature with two fully developed nervous systems and they join to become something of conjoined twins that can hold conversations with each other while being somewhat detached in what they sense and how they react but by the nature of a shared circulatory system and so would experience disease, hunger, and such much the same.
Perhaps the males and females become the dominant half when paired with a host. Think of this kind of like the host sex is a horse and the males and females are the rider that directs the horse where to go. Without a male or female to be the brains (as well as provide reproduction) the host will remain infantile and can only comprehend enough to eat, sleep, and soil itself.
The general idea is that all sexes are born as something infantile that need to feed off the "mother host" like an infant needs mother's milk. At some point the host sex develops a fully functional digestive system while the other two sexes remain small and dependent on a host. The male and female children could pair with a sibling host to fully mature so they can reproduce, perhaps the parents need to find an unrelated host for their male and female children, or maybe a child becomes so attached to the "mother host" that its competing with its mother for the host and could potentially replace the mother as the reproductive part of this pairing. The male and female sexes of this species would eventually develop a distaste for mother's milk and would have to attach to a host blood supply or die. If there's no alternative host provided then it could compete with a parent for its host, perhaps killing the parent in the process, somehow sharing the host, or lose the competition and die.
Because the host sex has the womb in this species than the female sex it could be that both the father (or rather sperm donor) can carry a pregnancy as easily as the mother (or rather egg donor). In that case it's not just the mother that is at risk of having a child attach to her host and kill her off but also the father is at risk, both can have hosts that can provide nutrition to developing children by a womb and milk.
I'm assuming a mammalian-like species with mention of mother's milk and wombs, with some thought I expect others could come up alternatives that are still a general variation on this theme.