In general, plastic surgery requires either 3 years of general surgery, complete ENT or Urology or Orthopedics or Neurosurgery residency. You can also do it with oral surgery as a dual degree with 2 years of general surgery.
Craniofacial surgery is a little different. There's a plastics side and an OMFS side. The Parkland program you're referring to is for people who complete plastics residency. There are also OMFS fellowships such as with Posnick (Washington, DC) or Costello (Pittsburgh). In my opinion, they are separated because the turf battle between the two specialties is one of the worst in health care. There are more plastics fellowships than OMFS (for craniofacial), but craniofacial surgery is more routine in OMFS with orthognathic & cleft surgery whereas few plastics guys do much more than primary clefts unless they are fellowship trained. I've also noticed trends in who performs which procedures. In many programs, it seems like plastics does more primary clefts whereas OMFS does more secondary clefts (bone grafting). My OMFS department sees other stuff like hemifacial microsomia and craniosynostosis, and I'm sure plastics here does also.