Ah ha! The parents
array is a list of objects with the id
field:
newfile = {'title': newtitle, 'parents' : [ { "id" : parentFolderId } ]}
service.files().copy(fileId=originalId, body=newfile).execute()
I'll update this if/when I figure out how to set permissions also.
One strange note here is that the file is still being copied to the drive root as well as the parent folder(s) I specify.