In my idiolect, a "parking space" is room for exactly one car to park, and a "parking lot" is a large piece of property with many parking spaces in it.
What's a "parking place"?
I think it probably just means the same as "Parking space" though if directly translated "Lle parcio" is just "Parking place" whereas "Parking space" would be "Gwagle parcio".
Right - but which of those two real-world concepts does the term refer to?
lle parcio is a parking space/place.
maes parcio is a car park or parking lot.
lle parcio gwag or gwagle parcio could be used if you are hunting for an empty parking spot/slot/bay/space.
In Welsh, lle can mean 'a space', 'one space, or 'some space'. The same with any noun, as there is no indefinite article 'a' in Welsh.
As with many languages, there is not always an exact one-to-one match between the range of meaning of a particular Welsh word or phrase and a given English one.