Why should you invite the Parkers if there is no cordial relationship, and actually quite the opposite (animosity to indifference). As you said, invite those who were there for you / your daughter.

Don't invite people you don't like or don't want to be there out of of a sense of what... obligation? duty? or to be 'nice'? Inviting them would not feel like a party but a chore and a burden & most certainly not fun. In that case, you may as well not invite any neighbours at all.

The party is to celebrate your daughter and have fun. So invite people accordingly.