The grammatically correct answer is "eats." The reason is that the subject of the sentence is the word "neither," which is singular. "Of my parents" isn't the subject, it's just a phrase modifying "neither." If someone said "We're inviting your parents to a party, can you tell me if they eat meat?" you could answer "Neither eats meat." Inserting the phrase "of my parents" doesn't change the grammar. The problem is that in situations like this, where there is a distance between the subject and the verb, we tend to automatically modify the verb to agree with the nearest word. It would be correct to say "my parents eat meat," and it is almost automatic to think and say, "parents eat," although grammatically the sentence is saying "neither eats." A native English speaker, when writing, would probably be careful, and use the correct form, "eats." But when speaking casually they might well say "eat" and nobody would even notice the mistake.

And my answer is "eats." Neither of my parents eats meat. I checked with the Cambridge Dictionary, which offers this explanation of why you might get conflicting answers from native English speakers: Spoken English: In formal styles, we use 'neither of' with a singular verb when it is the subject. However, in informal speaking, people often use plural verbs: Neither of my best friends was around. Neither of them were interested in going to university.

Eat

Eat

Eat