Eggs have an advantage over live births in that all they really need to gestate is a warm, safe place to be. The nutrients required for successful birth are already present in the egg.
If your oviparous species are females engaging in parthenogenesis (cloning themselves, without requiring a male to fertilise an egg), then Race A can be treated as literal incubators. Some hormonal changes may be required/desired in race A to allow for ingress of Race B’s eggs. Egg size matters. Fairy size eggs may not need anything special, baby size eggs may need quite a lot of work. Then ‘birth’ can either be of the egg just prior to hatching or of the hatched child and egg fragments. Either males or females of race A may be suitable candidates for this type of birth, depending on the... circumstances of the egg’s deposition. Sticky or hooked eggs can lodge in some truly unexpected places.
Needless to say some singularly squicky situations could arise here. Examples of this already exist in nature (Looking at you, spider wasps), though they generally result in the incubator being eaten alive by the children,
As for males of race B mating with race A, all I’ve got is that they’ve been preloaded with eggs (now sexually fertilised) much like a seahorse. In this case the B Male ovipositor is just used for depositing the eggs.
Obviously in this scenario the children are genetically only race B (and perhaps straight clones of their race B mother), but the member of race A may feel paternal/maternal instincts nonetheless.
If the babies of B/B couplings tend to kill their parents on birth/hatching and A’s are often sterile I can see this being a kind of social symbiosis evolving from a prior parasitic relationship. Sterile A’s get to have children, broody B’s get to not die in childbirth.