It isn't exactly a heavy, but its wide wings seem like it would be too big to fit in a medium parking spot, so where would it have been parked in real life?

I believe that it parked and flew as a "Heavy". 

At London Heathrow it was Stand 421

At New York JFK it was Terminal 7 Gates 5&6 (been awhile)

I'd have to pull up some old Concorde files if you wanted some more.

Wingspan wise was pretty small at just under 26m (smaller than a Dash8, or Embraer 170), but it was nearly 66m long.  You should be able to park at most any gates that can accommodate a 767 and larger.