Is there vehicle parking?? If not, where is the closest parking lot?

Yes. Small parking. Free. Limited parking right out in front of the cloisters.

Yes, there's free parking within the park. We parked outside the Cloisters.

There is parking

There's a parking lot at the Cloisters and one by the New Leaf Cafe, both within the park. They are small but free, I am fairly certain.