Neither do we need a parking area.

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.

Many adverbs which have a negative or "limiting" meaning cause SV inversion when they are at the beginning of the clause. Other examples : Never have I been so embarrassed; Not only did we lose our money, they also took all our credit cards; Hardly had I spoken when he walked in; We couldn't do it, and neither could the teacher!

The same happens with so: So badly did he fail, that he never wanted to try again; I like chocolate and so does David.

This is not a question form, just a second use of inversion (a third is "conditional "inversion - Had I known, I would have stayed at home! ).

The reason why need we doesn't work in that sentence is different. Need has two constructions. It can be a main verb - We need / Do we need/ we don't need - or a modal auxiliary - We need, we needn't, need we . But obviously, as a modal it can only be followed by another verb, not a noun. So, Neither need we cheese is not possible but Neither need we worry ... is fine : We needn't worry about cost and neither need we worry about maintenance.

A simple way to put this sentence would be:
We don't need wider roads or a parking area.

You could also say:
We don't need wider roads nor do we need a parking area.

Sorry, I sort of left this. Thanks for all of the replies.

I think I get it, now I just have to explain it 😮