otherwise I have to try whether I can manage a sketch.
You don’t have to. Just show all approved building documents where the parking spaces are clearly visible.
The building permit contains no details on the execution of the parking spaces.
Exactly that is now your problem. The building documents are too vague. I suspect your draftsman entered the parking spaces without elevation details in the site plan and/or ground floor plan. Thus, they are approved there, but if no changes in elevation are shown (are the street elevations given at all?), they cannot be regarded as approved. The building permit only covers what was applied for, plus any possible green entries from the authority.
That’s ridiculous, the excavator drove over it so often, nobody knows anymore what the original level was.
That confirms my suspicion regarding the simple building documents. In NRW you need a site plan with the existing elevations measured before construction activities.
Nevertheless, I see it like you:
However, I assume that the parking space, if it is indeed not exempt from procedures and its construction is not covered by the existing building permit, would still be approvable. That means I might have to obtain the building permit after all.
From your description of the matter, I see no rights of the neighbor violated, so nothing should oppose a possibly required subsequent approval. It just costs triple the fee.