There are separate Boards - the American Board of Anesthesiology for allopathic residencies, and the American Osteopathic Board of Anesthesiology.
Your legal privilege to practice anesthesiology will not be different coming from a DO or an MD residency. It might be easier to get a job in some cases with ABA certification than with AOBA cert in some cases, but if truly degree-ist hiring policies are in effect, you're screwed either way.