David,
As you probably know, Suffield Street is in that area which now includes Riverside Park and that there is also Durham Road, close by. I suspect that if there was a Durham Place, it was demolished long ago.
I think that somewhere in the dozens of boxes of paperwork which came from my mother's house when she died, there might be some maps of old Middlesbrough. If I can dig them out, I'll see if there is anything to help you.
Jen