"Roger Lynn" wrote
I didn't know that until 1974 they had their own county councils as
"Parts" of Lincolnshire, apparently in a similar arrangement to the
Ridings of Yorkshire.

And it seems that each of the county councils used the form 'parts' (plural)
in their full title within legal documents, e.g. "The County Council of the
County of Lincoln, Parts of Lindsey". Or so I'm told.