Cases in “Das sind meine Mutter und mein Vater”

Question

Mutter and Vater being the objects of the sentence, shouldn’t the preposition (?) mein for Vater change to meinen?

Meaning, why is it only correct to say:

Das ist meinen Tisch.

but not correct to say the following?

Das sind meine Mutter und meinen Vater.

Answer 1

It is not correct to say

Das ist meinen Tisch.

Correctly, it always has to be

Das ist mein Tisch.

Therefore, also the sentences

Das sind meine Mutter und mein Vater.

Das ist mein Vater.

Das ist meine Mutter.

are correct.

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Mein, meine, meiner [and so on] are possessive pronouns.

Kasus       Maskulinum     Femininum    Neutrum     Plural
Nominativ   meiner         meine        meins       meine
Akkusativ   meinen         meine        meins       meine
Dativ       meinem         meiner       meinem      meinen
Genitiv     meines         meiner       meines      meiner

Answer 2

As jera has explained, the correct sentences are

Das sind meine Mutter and mein Vater

and

Das ist mein Tisch

That said, I would like to add something to help you understand why.

I think you are getting confused with the accusative because you are following a simple (and unfortunately misleading) rule: if it answers the question "Who(m)? What?" then it is accusative.

For example, let's take the sentence "I eat an apple". You can think: "I eat what? An apple". So "an apple" must be the object, and thus it's accusative, so the translation in German is: "Ich esse einen Apfel". "Einen" is clearly accusative, and it is correct. But here comes the tricky part: this doesn't work with the verb "to be".

With respect to "This is a table", if you reason like this you will think "This is what?", and since "A table" is the answer, you will conclude it must be accusative. Unfortunately, it's wrong!

The golden rule is that with the verb "to be" (sein) you never have the accusative, it's always nominative.

Thus, "Mutter" and "Vater" in that sentence are not the objects. They are "predicative nominals".

Answer 3

You start from the assumption that the part after ist is the direct object of the sentence, but that assumption is wrong. Consequently, the conclusion that Mutter, Vater, Tisch should be accusative is also wrong.

One important role of the verb sein is that of the copula: It links the subject to a complement, which may be called subject complement, predicative complement or subject predicative. This complement, at least in German, is in the same case as the subject, i.e. in nominative.

Therefore, the only correct form of both sentences from the question is:

Das ist mein Tisch.
Das sind meine Mutter und mein Vater.

Canoo.net has a page with more details on complements (including subject complements and object complements).