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