Common marmoset (Callithrix jacchus) Wikipedia Commons
A news article in Nature (here) tells how new gene-editing techniques offer the opportunity to create transgenic primates. The emphasis is on neuroscience and models of brain disease. But if transgenic lines are developed there should be placentas available for study.
Although work on rhesus macaques is mentioned, more progress has been made using marmosets. One advantage might be that marmosets usually carry twins so a colony could be built up quicker. These Neotropical primates have placentas that differ in several ways from human placenta.
Haematopoietic foci in the placenta of a marmoset
(Callithrix jacchus) from Carter and Mess (here)
© Museum für Naturkunde Berlin
As an example, the placenta is an important site of haematopoiesis. The conventional wisdom is that there is little or no trophoblast invasion, but this is one aspect that calls for reexamination. If marmosets begin to emerge as important disease models there will be a need to look more closely at their placentation.