"Our people need water."
Translation:Naši lidé potřebují vodu.
I did write it's correct. But I don't believe it's commonly heard on the TV or radio (i.e. in literary speech). I certainly would not write "potřebujou" in any document or formal text.
As for Moravia, I guess I don't know the western part of it well. From my experience, people would say "potřebujú" in the Hodonín region, "potřebujó" in the Haná region, "potřebují" in Zlín and the northern part. Brno mostly has "potřebujou", so the western part probably does, too, as you say.
Yes, exactly. "Jedeš o víkendu k vašim?" (Are you going to your parents' place this weekend?)
Or: "Jak se mají vaši?" (How are your folks doing?)
And yes, "Naši potřebují vodu." would mean My parents need water. Or "our parents" if you have a brother/sister and he/she's nearby when you're saying the sentence - that's how this "naši/vaši" came to be.
Only "naši" and "vaši" works though. There's no such thing as "jejich" for the 3rd person, only the standard "jeho/její rodiče".