I finally broke down and bought a Henry Taylor narrow parting tool.
But looking at it, I don't have clue how to sharpen it. Can anybody
help? Thanks.

Woodboy writes:

I finally broke down and bought a Henry Taylor narrow parting tool.
But looking at it, I don't have clue how to sharpen it. Can anybody
help? Thanks.

From http://www.henrytaylortools.co.uk/parting.html

N.B. Fluted parting tools should be used flute down on the
rest. Sharpen by grinding the bevel only and maintaining the 45° bevel
angle.