Why does it sounds like an A instead of an Ä??? Is it a mistake or they pronounce it "Arzte"?
Why is seine translated as his here? It's female. How does one know when to use sei, sein, seinen, and seine?
seine is translated as "his" because that is what it means.
Be careful not to confuse the possessor and the possession.
seine is composed of two parts -- the stem sein- and the ending -e.
The stem sein- indicates that the possessor is male or at least grammatically masculine, i.e. "his".
The ending -e here shows that the possession is plural; it agrees with Eltern.
If it had been "her parents", it would have been ihre Eltern -- you would need a different stem ihr- to indicate a female or feminine possessor.
German is not like Romance languages in merging "his" and "her"; like English, it keeps them separate.
As for your second question: sei is a verb form: imperative ("be!") or subjunctive.
sein means "his" when the possession is masculine nominative, neuter nominative, or neuter accusative.
seinen means "his" when the possession is masculine accusative or dative plural.
seine means "his" when the possession is either feminine or plural, and either nominative or accusative.
For example, Seine Katze und sein Hund spielen mit seinen Pferden "his cat (fem.nom.) and his dog (masc.nom.) play with his horses (dat.pl.)".
Why is 'seine Eltern sind Ärzte', yet 'my sisters are doctors' is' meine Schwestern sind Ärztinnen'?
Ärztin is a female doctor.
Sisters are female, so if all of your sisters are doctors, they are all female doctors -- Ärztinnen.
Arzt is a male doctor; the plural Ärzte is also commonly used for mixed groups of male and female doctors. So his parents (presumably one male parent, one female parent) would be Ärzte.