Don't be thrown off the tracks anymore, here´s a hint: the answer, it it is going to be about "them" (thus, it´s plural) so the question use DO.
If the answer is about "he/she/it" then the question uses DOES.
DO your cats drink milk?
Who drinks milk? The cats (they). Perhaps you were thrown off by that (latter) question
milk, but the trick is to focus on the answer THE CATS.
DO your parents live with you?
Who lives with you? Your parents. So, we are talking about "THEY" (your parents, not only one - even if you have only one at the time, but the question is general in kind).
DOES your parent live with you, or separately?
Here the asker really knows that you have only one parent (alive perhaps, or only one has raised you).
The answer is "he/she/one
of them" lives with you. Hence the DOES in the question.
Is a case of "the answer is in the question" and vice versa.
Thus, back to the crux of the matter:
"does your cat drink milk?"
Note that the use of DO/DOES/DID, etc, "kills" that thing of using "S" when singular - but then it´s signaled by the use of DOES in the third person singular (he/she/it). And the expected answer could be indeed "he/she/it" drinks...
The word CAT also is an indication.
"do your cats drink milk?"
It´s not about the third person singular, so with present tense here, we use DO.
CATS is an indication of that as well (CATS being a plural word).