Using "Have" and "Has" with "Or"

Hi,

I am having trouble understanding which one of the following sentences is correct:
1. Either John or Mary has eaten some cake.
2. Either John or Mary have eaten some cake.
3. Either John or your parents have eaten some cake.
4. Either your parents or John has eaten some cake.

I believe only sentence 2 is incorrect as "have" must agree with the singular noun Mary.
Can anyone confirm for me that the "have" and "has" in sentences 3 and 4 are correct? I believe they need to comply with the noun closest to them but am having trouble confirming this.

Thanks.

H.