because that is the feminine form of "ihr".

quick summary:

der Hund / die Katze / das Haustier

Singular:

1. Person: mein Hund / meine Katze / mein Haustier
2. Person: dein Hund / deine Katze / dein Haustier
3. Person:
   feminin: ihr Hund / ihre Katze / ihr Haustier maskulin: sein Hund / seine Katze / sein Haustier

Plural:

1. Person: unser Hund /unsere Katze / unser Haustier
2. Person: euer Hund / eure Katze / euer Haustier
3. Person: ihr Hund / ihre Katze / ihr Haustier

You could use the turtle. : )

Ihr can have multiple meanings but I believe the difference in this case is because of the grammatical gender of the words.

Mutter is feminine and Vater is masculine, hence ihre and ihr, eine and ein, die and der.

Ihre is also used for plurals, e.g. ihre Jungen.

That translation is fine.

Exactly, i think that maybe mother and father is threatened in this case as "possessive", in possessive pronouns Sie turns into Ihr

When it means "your" (belonging to the polite "you"), it's always capitalised, even in the middle of a sentence: Ihr.

"her" or "their" is lowercase, ihr -- except of course at the beginning of a sentence. There is absolutely no way to tell the difference between "her" or "their", except for context.

Because then it would have been seine Mutter und sein Vater.

The ihr- shows that it is “her” or “their” mother and father.

Is the "ihr" required here?

Yes.

is it because the genders of Mutter and Vater differ

That's definitely one of the reasons it sounds wrong to me.

could it be "Ihre Mutter und Schwester"?

That sounds to me like "the person who is both her mother and her sister".

I wouldn't completely rule out the possibility of using it for "two people: (a) her mother, (b) her sister". But if you want to be clear, I would use the possessive determiner twice.