The definition of a sex is that each sex produces a unique type of gamete. If it does not produce unique gametes then it is not a sex. A caregiver that does not contribute genes is a caste or morph, not a sex.
The advantage of being triploid and trisexual is that this reduces the risk of genetic bottlenecks and inbreeding depression.
The disadvantages are that three individuals are required to reproduce and, contrary to popular belief, reduce phenotypical diversity. Recessive alleles are fair less likely to display phenotypes because three copies would be required.
An environment where trisexual species outcompete dioecious species would need to place greater selection pressure on genetic redundancy over the costs of finding mates and reduced phenotypic diversity.
An example of a true trisexual species may be found @ sff.net/people/dragonwriter/worldbuilding/trisexual.html
EDIT: the ant genus Pogonomyrmex actually has four sexes on the demographic level. Each colony requires three individuals to reproduce: the female mates with males of her species and another. Males are spawned asexually by females. Same species mating produces only reproductive females. Different species mating produces sterile hybrid females that form the worker caste. Collectively all four sexes, two males and two females of different species, are required for both species to survive.
To put this by analogy with cells in one organism: reproductive cells require gametes from the other species to produce somatic cells.
This would not work the same way on the organism level. There the analogy would be the female produces a sterile hybrid offspring which she parasitizes similar to the anglerfish.