I've done it both ways.
If you have a kit from Berea there is a ring that goes onve the step bushing where the CB will go.
In that case it is easier to part first and finish.
With Woodcraft kits on the American Flat top that ring is not supplied.
In that case I finish first,sharpen my skew and parting tool, say a prayer to the Pen Gods, nad proceed slowly but deliberately.
If you part first then finish,you may need to take a skew to clean up the tube,especially with a CA finish.