Neither takes an inverted word order, as in: neither + auxiliary verb + subject. So we would say, for example, "I don't like cheese. Neither do I." In the case here, need in the negative uses do, as in: "I don't need any more cheese, thank you!" But combine the two grammar points, and the word order gets all funky... kind of like cheese.
"Neither need we a parking area" is grammatically wrong, whereas your sentence is correct, "Neither do we need a parking area."
Don't ask me what my deal with cheese is, by the way.